What would a world where $\mathsf{CH}$ is false look like?

If we assume the axiom of choice then we can construct a set of every possible cardinality. This is easy because cardinalities are really just particular sizes of ordinals, so we just need to go to the particular ordinal of interest and we have a set of the wanted cardinality.

What is, however, much harder is to point out a set of size $\aleph_1$ of real numbers. This sort of trickery requires the axiom of choice, and in fact, once one is versed enough in this sort of idea, it becomes quite easy. We can construct a surjection from $\Bbb R$ onto an ordinal of size $\aleph_1$, and using the axiom of choice we can construct an injective inverse whose range is such a set.

But, comes the million dollar question, can we write down an injection from the real numbers into an ordinal of size $\aleph_1$? And this, in fact, is the continuum hypothesis.

Assuming the continuum hypothesis holds, or fails, doesn't change our construction of the real numbers, nor it changes our construction of the ordinals. It only changes the answer to the above question. Assuming the continuum hypothesis the answer is positive, and assuming it fails the answer is negative.

It should also pointed out that the negation of equality is merely inequality. So if the continuum hypothesis fails, we did not yet decide the exactly cardinality of the continuum. It might be $\aleph_2$ or it might be $\aleph_{\omega_1}$. If we do know what is the cardinality of the continuum, then by definition we know that there is a bijection between the real numbers and a particular ordinal, restrict it to meet only certain (smaller) ordinals would result in sets of real numbers whose cardinality is between the natural numbers and the real numbers.

But if we don't know the cardinality of the continuum, just that it is not $\aleph_1$, then we can't say much more than "There is a set of real numbers of size $\aleph_1$, and the real numbers themselves are not a set of size $\aleph_1$". However, their construction from Dedekind cut would be exactly the same.

As tomasz suggests in the comments, we can point at a few values which are impossible for the continuum. The first cardinal which has infinitely many infinite cardinals smaller than itself, also known as $\aleph_\omega$, cannot be the cardinality of the continuum. This is due to Koenig's theorem about cardinal arithmetics. The other point of interest is that Cohen proved that the continuum is consistently $\aleph_2$, but this result was extended by Solovay and it was shown that pretty much any value which is not forbidden by Koenig's theorem is consistently the cardinality of the continuum.

I should say a word about the axiom of choice, that without it it might be the case that the real numbers might not be equipotent with an ordinal to begin with, and it might be that there is no set of size $\aleph_1$ of real numbers; it still might be the case that there are intermediate cardinalities between the real numbers and the natural numbers, and it is possible that there are aren't any. But without the axiom of choice a lot of things become harder to work out anyway. Not the construction of the real numbers, though. That's just as simple as with the axiom of choice.