Measure the height and width of the closet's back wall with a tape measure.
Cut two 2-by-4-inch boards to match the height of the closet's back wall. Apply construction adhesive to one of the 4-inch sides on each board. Place the boards onto the side walls of closet with the construction adhesive pressing against the wall and the back edges of the boards 4 inches from the back wall of the closet. Let the adhesive dry thoroughly.
Cut a piece of 1/2-inch thick plywood to match the height and width of the back wall of the closet. Place the board against the 2-by-4-inch boards. Drill a 1/4-inch hole on each corner of the plywood, drilling through the plywood and into the 2-by-4-inch boards.
Place wood screws into each of the holes and tighten them to hold the false wall in place.
Paint the wall to match the color of the closet interior.