Is there a shape that can be wrapped perfectly? Wrapping presents in the real world always involves overlapping paper (due to folds, etc).
Is there any shape that can (theoretically) be wrapped by a rectangular piece of paper without any overlap (the shape and the paper have the same surface area)?
If such a thing exists, I imagine it would have to have angles to allow the paper to wrap to another side.  I don't care if the shape is concave or convex.
The shape must have a volume greater than 0

One solution is a regular tetrahedron. We can even generalize this to tetrahedrons constructed from regular ones where we just pull two oposing edges apart. The following pictures show a 3D models in blender. The red edges show where we cut the surface apart (called seams) and on the right side we see the unwrapped net of each model (done using UV-unwrapping, usually done for texturing objects). We need to cut one triangle in half in order to get a rectangle (otherwise we'd just get a parallelogram). 
We can easily observe that this technique can be used for any side ratio of rectangles.




