Cut the plywood into 2-by-2-foot squares. You will need six of these. Use a saw and measure carefully before you cut, because your box must be uniform in order to be sturdy.
Lightly sand the sawed edges so they will fit together snugly when you construct your box.
Select one square for the top of your box and set it aside. Stand a second square on its edge and place a bead of wood glue on the top edge this square.
Place the top square on the second square that is standing on edge. Align the two edges so that the top square rests on the support square.
Nail through the top of the square and into the supporting square.
Repeat steps 3 through 6 for the remaining three support squares. You will end up with a five-sided box.
Turn your five-sided box over. Cut the 2-by-2-inch lumber into lengths that will fit inside the box when placed along the seams.
Glue the lumber pieces in place along the seams to support the box's joints. Place them snugly along the seams and be sure to glue all the surfaces that touch the plywood.
Place a bead of glue on all the exposed edges of the five-sided box. Place the bead away from the edges. If it soaks into the wood, apply a second time.
Attach the final square by aligning it with all the edges. Nail it down. Let your structure dry overnight.
Turn your box over. Estimate where the 2-by-2-inch supports are glued. Drive nails through the plywood and into the supports underneath. This will add stability to your step box.
Cover all nail holes with plastic wood. Allow it to dry, then sand it down so that it is level with the wood of the box.