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Before starting this project, cut away the drywall under the studs and clean out the space. Carefully measure the space (Image 1) and sketch out a design for the home office (Image 2). One idea is to design the office inside a giant box for easier installation and construction.
Cut out the pieces for the office "box" from 3/4" birch plywood. If one side of the office box is angled, mitere the edges of the plywood at those joints for a better fit. Also cut pieces (including ledger strips) for the office desk and shelves, according to the design. Always wear safety glasses when working with a power saw or any other power tool.
Assemble the office box using 2" wood screws. Drill pilot holes using a countersink bit before driving the screws into the wood. Don't use wood glue on the pieces yet -- they will be taken apart and re-assembled in the office space.
Once the box is assembled and all joints fit properly, carefully back out the screws and disassemble the office. Sand and stain the box pieces, then apply several coats of polyurethane sealer to protect the wood. Also stain and finish the pieces for the desk and shelves. Why build the box just to take it apart? Pre-assembling the box shows whether the pieces fit properly before they're stained, finished and moved indoors for installation.
Cut a piece of pressure-treated, 3/4" plywood to fit the floor under the office, then use subfloor adhesive to adhere it to the floor. This brings the floor under the stairs level with the rest of the room. Moisture from the earth can leach in through concrete and rot out wood. If wood will come in direct concrete with concrete, always use pressure-treated wood to combat rot.
Bring the box pieces to the office area and re-assemble the box using screws and wood glue. The assembly should go much faster this time, since the screws go back into their original holes. Place the bottom of the office box on blocks of 2" x 4" lumber for easier assembly and movement.
Slide the box into the cavity under the stairs, lining it up with the angled cut in the drywall. Use screws to secure it to the studs in the back wall.
Install the finished ledger strips in the box: Strips on the back and right wall will support the desk; a strip on the left wall will support the shelf.
The desktop and shelf are supported by the ledger strips and by a vertical support. Attach the desktop and shelf to the support, then attach the entire structure to the office box and ledger strips. Tack these pieces in place with a nail gun to hold them steady, then install screws for strong support.
Nail 3/4" decorative trim to finish the edges of the desk. Cut and install wider wood trim around the office opening to give a finished look and cover any gaps between the office box and the surrounding wall.