The four pieces don't have measure, don't have unambiguous volume. It might help to think of them as fractals (though in fact they're even more ambiguous than that).
The only "realistic" answer I can give is that each of the four pieces would "take up" the full volume of the sphere as you moved them apart. Each of the four pieces is "shaped" like the whole sphere, but with infinitesimally small holes where the other four go. So not a single "water droplet" would ever go inside the four pieces, and while you were moving them around they'd take up the space of four of the original sphere.
How are they more ambiguous than fractals? The key structure is a shape that can be rotated onto a subset of itself.
I guess with common fractals, you camnoly scale an object onto a subset of itself. The magic of BT is that the ball-shape allows a rotation that looks like a scaling.
A fractal tends to be kind of contiguous - for most points in the fractal there's a neighbourhood of those points that's also in the fractal, and similarly for points not in the fractal. These are "dust sets", dense everywhere in the sphere and their complements also dense.
Fractals generally have a formula describing them, i.e. they are computable. Nonmeasurable sets are necessarily non-computable. This is why you will never see an accurate picture of the B-T sets, because they are not defined uniquely by any formula or algorithm (which is connected to the fact that the axiom of choice is needed to even prove these sets exist).
The only "realistic" answer I can give is that each of the four pieces would "take up" the full volume of the sphere as you moved them apart. Each of the four pieces is "shaped" like the whole sphere, but with infinitesimally small holes where the other four go. So not a single "water droplet" would ever go inside the four pieces, and while you were moving them around they'd take up the space of four of the original sphere.