It's obviously possible to dissect a sphere into pieces that could be reassembled into something that looks like a cube from the outside. Inside, it would be hollow, possibly with a bunch of spare pieces rattling around inside. Feasibility sketch: build thin polyhedral planks from the inner part of the sphere, with 45-degree bevels where necessary to let them form the cube's edges and vertices. All the extra material could be chopped up into manageable chunks and hidden in the interior. If there's too much extra material, redesign with thinner planks to make the outer cube bigger. Intuitively, it feels like we could get away with just a few dozen pieces or so: maybe 4 to 6 per face, and then an approximately equal number to pack inside ... but I don't know. Can anybody provide an explicit construction, or make lower-bound arguments about the number of pieces?