Cut the plywood to the size of your mattress. These instructions are written for a standard queen size mattress, which is 60 inches wide and 80 inches long.
Cut four pieces of the 2-by-4 to make a square frame that attaches to the bottom side of the plywood. This requires two pieces 80 inches long and two pieces 57 inches long. Note that the actual dimensions of a 2-by-4 are 1 1/2 inches by 3 1/2 inches.
Drive screws through the 80-inch 2-by-4s and into the ends of the 57-inch 2-by-4s to create a rectangular shape. The 3 1/2 inch dimension of the 2-by-4s should be vertical, and the 1 1/2 inch dimension should be horizontal. The outside dimensions of this rectangle will be 60 inches by 80 inches, the same as the piece of plywood.
Position the piece of plywood on top of the 60-by-80 rectangle and drive screws down through the plywood and into the 2-by-4s. A screw every 12 inches will make the construction sufficiently strong.
Turn the construction over so the plywood is on the bottom. Fit three more 57-inch pieces of 2-by-4s in between the original end pieces, spaced at 2-foot intervals. This will make the platform much stronger for supporting the mattress. Screw them in by driving screws through the outside face of the 80-inch side pieces and into the ends of the 57-inch pieces.
Cut four pieces of the 2-by-4s to a length that equals the distance you want your mattress from the floor, minus the thickness of the mattress and the 3/4-inch piece of plywood.
Fit one of these pieces of 2-by-4 into each corner of your construction so that the outside faces of the 2-by-4s press against the inside corners created by the 57-inch 2-by-4s and the 80-inch 2-by-4s. You should now have a platform sitting upside down on the floor with four legs pointing vertically out of it.
Screw the legs onto the construction by driving screws through the outside faces of the 57- and 80-inch 2-by-4s and into the 2-by-4s that make the legs. Driving screws in from two directions will make the legs stronger.
Turn the entire construction over so that it is standing on its legs. Place the mattress on top of it.