Measure and cut each closet cube from four boards that are 11 1/4 inches from the 1-by-12-inch boards. These will be the sides of the cubes.
Assemble the cube by creating a square with offset corners. Place the first side on its narrow edge, perpendicular to a flat work surface. Apply wood glue to one edge of a second side.
Lay the second side flat on the work surface next to the first side, with the wood glue facing the first side, which is standing on its narrow edge. Push the wood sides together and line up the edges. Secure the corner with three wood screws that are 1 1/2-inches. Repeat with two more pieces so you have two L-shaped halves of a cube.
Flip over the second L made from two sides of the cube. Position it on top of the first L so a rectangular cube is formed with an open front and back. Once you have the second L positioned properly, move it aside and apply glue to the two edges that touch.
Put the second L back in the right position, line up all the edges and secure the two corners with screws.
Cut one piece measuring 12 1/2-by-12 1/2-inches from the hardboard sheet for the back of the cube. Turn the cube so the front open end of the cube is facing the work surface. Place the hardboard square you cut on the back of the cube, align all the edges and secure the back to the cube using 1/2-inch finishing nails.
Sand any rough edges first with 100-grit sandpaper and finish sanding with 180-grit sandpaper.