Cut four pieces from three-quarter-inch sheet lumber that are 14 inches by 15 3/4 inches and one that is 16 1/2 inches square. Cut a 2-by-2 board into four 16-inch-long pieces.
Build a wooden box from the five pieces of three-quarter-inch lumber, using the largest piece as the bottom and the four identical pieces as the sides. Assemble the pieces with wood glue and then secure them with nails along each seam.
Place one 2-by-2-by-16-inch leg into each corner of the box so that they stand 1 inch higher than the box edge. These will be the legs. Glue and nail each board into place.
Turn the box over so that the opening faces downward. Cover each flat surface of the cube with a piece of sheet batting cut to size, wrapping the edges underneath the cube. Staple the batting into place and trim away the excess.
Cut four squares of upholstery fabric that are 14 1/2 inches by 16 1/4 inches and one that is 17 inches square. Sew one of the four identical pieces to each edge of the largest piece with the right sides together using a one-half-inch seam allowance. Turn the piece wrong-side-out and stitch down each corner. This will serve as the slipcover.
Trim away the extra thread and turn the slipcover right-side-out. Pull the cover down over the padded cube. Place your cube in the desired location.