The following blocky shape:
can be folded onto the surface of a cube in a way that perfectly covers the entire cube with no gaps and no overlaps.
How can it be done?
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This seems to work:
Below, I printed out the shape, and cut off the excess. The white parts are for glueing; if everything works out as planned, all of them will be covered by the coloured bits around the black squares.
Joy, it all worked! Here's the final cube, with some white "intentionally" showing through between the pieces, highlighting the borders:
The key to solving this was to
notice that the big squares are way too big to be the side of the cube. Also, because the big squares need to be folded, and the cube sides are going to be smaller than them, all the 4 corners of one cube side really want to be touching the edges of the big square: if you were to place a 3D corner anywhere inside a solid area, the folds would unavoidably cause a wrinkle.
After that, it was a matter of figuring out the proper size and orientation of the cube's edge that's inside the big square. I first tried "rotated 45 degrees", but that didn't seem to pack well. The natural next guess was to put a fold through the 270 degree "inside" angle, which fixed the cube side's position inside the big square, and that happened to work all the way through.
The shape can be folded like this
Then it should become obvious.
Method of solving:
I started out by cutting out the shape, noticed some equal lengths, and initially folded it in the wrong places (as can be detected by creases in my model). I then wondered if the four smaller rectangles would wrap around four sides of the cube. I marked the parallel diagonals of the four small rectangles, and noticed that a line touching the tips of their saw-tooth edges is at 90° to them. I could then mark out the two other faces in the largest squares, and from there it all fell into place, leaving the larger squares as the top and bottom caps, folded to fill in the gaps.
As I was working with a print of the black image in the question, it proved rather difficult getting a photogenic shot, and I ended up retouching it to show where the original edges are.