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Working on one side at a time, glue and nail the side to the back. Apply glue and drive three 1-5/8-in. nails into each shelf, attach the other side and nail those shelves into place to secure them. Clamps are helpful to hold the unit together while you’re driving nails. Center the top piece, leaving a 2-in. overhang on both sides, and glue and nail it into place. Paint or stain the unit and then drill pilot holes into the top face of each side of the unit and screw in the hooks to hold your ironing board. Mount the shelf on drywall using screw-in wall anchors.
Start with ¾" plywood that is twice as wide as the depth of the cabinet and as long as one side and top/bottom. I crosscut the top and bottom off of one end and set it aside. Then, using a ½" dado stack in my table saw, I cut a series of ¼" deep dadoes across the sides. Make the spacing between them uniform so that the drawers are interchangeable. Finally, rip the sides and top/bottom in half to separate them and assemble the cabinet however you please. If you want to hang it on a wall, attach a French cleat to the back at the top and a batten of equal thickness at the bottom.