Can the Hubble constant be measured locally?

Everything doesn't expand equally because of cosmological expansion. If everything expanded by the same percentage per year, then all our rulers and other distance-measuring devices would expand, and we wouldn't be able to detect any expansion at all. Actually, general relativity predicts that cosmological expansion has very little effect on objects that are small and strongly bound. Expansion is too weak an effect to detect at any scale below that of distant galaxies.

Cooperstock et al. have estimated the effect for systems of interest such as the solar system. For example, the predicted general-relativistic effect on the radius of the earth's orbit since the time of the dinosaurs is calculated to be about as big as the diameter of an atomic nucleus; if the earth's orbit had expanded according to the cosmological scaling function $a(t)$, the effect would have been millions of kilometers.

To see why the solar-system effect is so small, let's consider how it can depend on $a(t)$. There is a cosmology called the Milne universe, which is just flat, empty spacetime described in silly coordinates; $a(t)$ is chosen to grow at a steady rate, but this has no physical significance, since there is no matter that has to expand like this. The Milne universe has $\dot{a}\ne 0$, i.e., a nonvanishing value of the Hubble constant $H_o$. This shows that we should not expect any expansion of the solar system due to $\dot{a}\ne 0$. The lowest-order effect requires $\ddot{a}\ne 0$.

For two test particles released at a distance $\mathbf{r}$ from one another in an FRW spacetime, their relative acceleration is given by $(\ddot{a}/a)\mathbf{r}$. The factor $\ddot{a}/a$ is on the order of the inverse square of the age of the universe, i.e., $H_o^2\sim 10^{-35}$ s$^{-2}$. The smallness of this number implies that the relative acceleration is very small. Within the solar system, for example, such an effect is swamped by the much larger accelerations due to Newtonian gravitational interactions.

It is also not necessarily true that the existence of an anomalous acceleration leads to the expansion of circular orbits over time. An anomalous acceleration $(\ddot{a}/a)\mathbf{r}$ just acts like a slight repulsive force, which is equivalent to reducing the strength of the gravitational attraction by some small amount. The actual trend in the radius of the orbit over time, called the secular trend, is proportional to $(d/dt)(\ddot{a}/a)$, and this vanishes, for example, in a cosmology dominated by dark energy, where $\ddot{a}/a$ is constant. Thus the nonzero (but undetectably small) effect estimated by Cooperstock et al. for the solar system is a measure of the extent to which the universe is not yet dominated by dark energy.

The sign of the effect can be found from the Friedmann equations. Assume that dark energy is describable by a cosmological constant $\Lambda$, and that the pressure is negligible compared to $\Lambda$ and to the mass-energy density $\rho$. Then differentiation of the Friedmann acceleration equation gives $(d/dt)(\ddot{a}/a)\propto\dot{\rho}$, with a negative constant of proportionality. Since $\rho$ is currently decreasing, the secular trend is currently an increase in the size of gravitationally bound systems. For a circular orbit of radius $r$, a straightforward calculation (see my presentation here, sec. 8.2) shows that the secular trend is $\dot{r}/r=\omega^{-2}(d/dt)(\ddot{a}/a)$. This produces the undetectably small effect on the solar system referred to above.

In "Big Rip" cosmologies, $\ddot{a}/a$ blows up to infinity at some finite time, so cosmological expansion tears apart all matter at progressively smaller and smaller scales.

Cooperstock, Faraoni, and Vollick, "The influence of the cosmological expansion on local systems,"

We do measure the Hubble constant locally: everything that we know about it comes from observations of light in the vicinity of our telescopes. But if you restrict the experiments to a room with opaque walls, then no, it can't be measured locally, because it merely quantifies the average motion of galaxies on large scales, and there's nothing in the room that will tell you that. Note that earlier answers to this question, including the accepted answer, are wrong, inasmuch as they all suggest that it could be measured locally in principle if not in practice. The paper by Cooperstock et al is also wrong.

The error in Cooperstock et al is easy to explain. They assume that the cosmological FLRW metric is accurate at solar system scales. You can plug the FLRW metric into the Einstein field equations (or the Friedmann equations, which are the Einstein equations specialized to FLRW geometries) to see what this implies about the stress-energy tensor. What you'll find is that they've assumed that the solar system is filled uniformly with matter of a certain density and pressure. The force that they calculate is simply the local gravitational effect of the matter that they assumed was present. But it isn't actually there. It's elsewhere: it collapsed into stars and planets. When they treat the cosmological force as a perturbation on top of the usual solar-system forces, they're double-counting all of the matter, once at its actual location and once at the location where it hypothetically would be if it hadn't clumped. Matter only exerts a gravitational influence from its actual location.

General relativity is different from Newtonian gravity, but it's not as different as many people seem to imagine. It's still a theory of gravity: a force between massive objects that's mediated by a field. It's not a theory of test particles following geodesics on meaningless spacetime backgrounds. The FLRW geometry is not a background; it's the gravitational field of a uniform matter distribution. It could be roughly described as a bunch of Schwarzschild patches stitched together and then smoothed. In real life, there is no smoothing, and no FLRW geometry; there is only the (approximately) Schwarzschild local patches. There is no universal scale factor evolving to the ticks of the absolute, true and cosmological time; there is only local motion of ordinary gravitating objects. That this averages out, on huge scales, to a FLRW-like shape with local bumps is known to us, but irrelevant to nature, which only applies local physical laws independently in each spacetime neighborhood.

Measuring the Hubble constant in a sealed room is no different from measuring the abundance of helium in a sealed room. It will only tell you what's in the room. The abundance in the room won't tend to 25% over time. There's no subtle residual effect of 25% abundance that you can measure locally. The universe is about 25% helium because most of the helium from the first three minutes is still around, not because there's a local physical process that regulates the amount of helium.

What about dark energy? Dark energy, by assumption, doesn't clump at all. You can measure its gravitational effect in the room because it's present in the room. The acceleration you'll measure is not $\ddot a/a$, because $\ddot a/a$ incorporates the averaged effect of all matter, not just the stuff in the room. In the distant future, as $Ω_Λ$ approaches $1$, the acceleration you measure will approach $\ddot a/a = H^2$, but there's no way for you to know that unless you look outside of the room and observe that there's nothing else out there. If dark energy clumps (by little enough to evade current experimental limits) then the amount in the room may be smaller or larger than the average. In that case you'll measure the effect of what's actually in the room, not the effect of the average that you're taking it to be a perturbation of. Nature doesn't do perturbation theory.

Same with dark matter. There may be some of it in the room, depending on what it's made of. If there is, the density will probably be larger than the universal average, but it could be smaller, or about equal. In any case, what you'll measure is what's actually in the room, not what would be there if dark matter didn't clump.

Here are some comments on specific parts of other answers.

For two test particles released at a distance $\mathbf{r}$ from one another in an FRW spacetime, their relative acceleration is given by $(\ddot{a}/a)\mathbf{r}$.

That's correct. Assuming the F(L)RW geometry in GR is equivalent to assuming a $(\ddot{a}/a)\mathbf{r}$ field, or $(\ddot{a}/a)\mathbf{r}^2/2$ potential, in Newtonian gravity. By Poisson's equation that implies uniform matter of density $\ddot{a}/a = -\tfrac43 πGρ$ is present everywhere.

Within the solar system, for example, such an effect is swamped by the much larger accelerations due to Newtonian gravitational interactions.

That's incorrect. The effect is absent in the solar system because the matter that would have caused it is absent. This is obviously true in Newtonian gravitation; it's also true in GR.

The actual trend in the radius of the orbit over time, called the secular trend, is proportional to $(d/dt)(\ddot{a}/a)$

I think that this would be correct if the $(\ddot{a}/a)\mathbf{r}$ force actually existed.

Note, though, that if the force existed, it would be due to, and proportional to, the mass located inside the orbital radius, so you may as well say that the trend is proportional to $dM/dt$. This holds regardless of the nature of the mass; it could be a star losing mass to solar wind and radiation, for example. For a circular orbit $mv^2/r=GMm/r^2$, which gives $dr = d(GM)$ if you hold $v$ constant, so this seems reasonable.

If you added FLRW matter to the solar system, you wouldn't get this trend, because it would clump on much smaller time scales. To follow the Hubble expansion over long time scales it would have to behave totally unphysically: gravitationally influencing other matter but entirely uninfluenced by it, just sedately expanding independent of everything else. This happens when the FLRW matter is the only matter in the universe, since there's nothing to break the symmetry; otherwise it makes no sense.

if you write down the Einstein equation for the case of a simple cosmological-constant dominated universe and a spherically symmetric matter source [...] you [...] get an instability in orbits whose radius is greater than some value $r_∗$, which is proportional to $1/(ΛM)$. This outermost instability represents the expansion of the universe starting to dominate over objects orbiting very far from the star [...].

It represents the dark energy, which is present locally, starting to dominate. As you go to larger radii, the total contained dark energy goes up roughly as $r^3$, and $r_*$ is the radius at which the repulsive force from that equals the attractive force of the central mass. Mass outside that radius can be neglected by the shell theorem/Birkhoff's theorem. This doesn't tell you the Hubble constant or the scale factor; it only tells you the local density of dark energy, which as I mentioned before can be measured inside the opaque room.

Ben Crowell's answer is right, but I am adding a point in order to emphasize it, because this issue keeps coming up. Here is the point:

The cosmological expansion is FREE FALL motion.

What this means is that the clusters of galaxies on the largest scales are just moving freely. They are in the type of motion called 'free fall'. It means they are going along, with their velocity evolving according to whatever the net average gravity of the cosmos as a whole says. There is no "inexorable space expansion force" or anything like that. They are not being carried along on some cosmic equivalent of tectonic plates. They are just falling. In technical language, their worldlines are geodesic. This should help you to understand why forces within galaxies, and within ordinary bodies, will hold those galaxies and those bodies together in the normal way. It is not essentially different from objects falling to Earth under the local gravity: Earth's gravity offers a tiny stretching/squeezing effect, but this is utterly negligible compared to all the ordinary forces within materials.

If you could somehow switch off the gravitational attraction within the solar system and the galaxy and the local cluster, and all the electromagnetic and other forces, then, and only then, would the parts of the solar system begin to drift apart under cosmic free fall motion, commonly called the expansion of space.