Volume of regular octahedron

The volume of a pyramid is $\frac 13 hA$ where $h$ is the height and $A$ is the area of the base. Using this for two pyramids suck together gives me $\frac{\sqrt 2}3\approx 0.47$ for a regular octagon of side $1$.

You are implicitly arguing that the volume of a pyramid is $\frac 12 hA$ - I think you are applying two dimensional intuition to three dimensions. The area of a triangle is $\frac 12 bh$ because the similar pieces diminish linearly if you draw lines parallel to the base. But the areas of the cross sections of the pyramid here are two-dimensional.