OK, I'm not very clever, but I think I'll take a shot at this. Center the box on a square piece of paper. The distance from the center of the paper/box to the edge of the paper needs to be 2 meters (.5 for the bottom, 1 for the side, and .5 for the top) so the pieces meet on the opposite side of the box. Any less and it'll fall short. The way to minimize paper use is to rotate the box 45 degrees so it's "diamond shaped" on the square piece of paper. That way the required 2 meter length will be on the diagonal of the wrapping paper, the longest distance from centre to edge of the paper.
So the entire diagonal will be 4 meters, each side is sqrt(8)....
So, 8 square meters of paper? One square of paper the square root of 8 on each side? (Square square square.)
But geometry is not my long suit. Anyone out there know a better way to do this with some creative folding?
(Edit: Alternately, if it was a 1x8 sheet, you could wrap 4 sides of it with the middle 4m and then do 45 degree folds to cover the last two sides with the ends, or wrap 4 sides starting at one end, then do a 45 degree fold and wrap the last two sides with the last 3m.)
Also, I like the way this d6 I wrapped up in a square of toilet paper looks.