I can wrap a unit cube with a square of diagonal 4 (side thus 2sqrt(2)). I think there is a smaller quadrilateral that will do it, but I can't prove that. Be Well! Dane Brooke

Joseph DeVincentis -- Wrapping a cube: a square of area 8 times a cube-face works if you position the cube right in the middle of the square but rotated 45 degrees with respect to the square, and then wrap the paper up around each side of the cube, folding section over at each of the four vertical edges, but there's probably a better way.

I don't know if it's the optimal answer, but I can get an 8 square-inch square to fold into a cube:

The white parts are what faces out, the red parts are pinched together and folded under the surface. The four triangles in the corners come together to form one face.