Lay a plank of 50-inch 1-by-10 birch lumber flat on your work surface. Apply wood glue to the 1-inch edges along the 50-inch side.
Abut a piece of 50-inch 1-by-10 birch lumber on either side of the glued one, and press them together, so that the glued edges stick to the edges of the outer two planks. This will create the desktop, measuring 30-by-50 inches.
Clamp the three desktop planks together with F-clamps, ensuring the planks stay flat as they are clamped. Let the glue dry.
Glue a sheet of 20-by-40-inch 1/2-inch birch plywood over the underside of the desktop planks, centered. Secure the plywood to the underside of the desktop with 1 1/4-inch wood screws every 5 to 6 inches around the edge of the plywood, and on 6-inch centers across the face of the plywood.
Lay a piece of 20-by-40-inch 1/2-inch birch plywood flat on the work surface. Glue a plank of 40-inch 1-by-6 birch lumber upright along one of the 40-inch edges of the plywood, so that the bottom of the plank is flush with the bottom of the plywood, and the plank is perpendicular to the plywood.
Glue two planks of 21-inch 1-by-6 lumber upright against the short sides of the plywood. These side planks should be flush ends of the 40-inch plank of 1-by-6 lumber, to create a square “U” shape around three sides of the plywood.
Secure the planks to the plywood with 1 3/2-inch wood screws driven every 5 to 6 inches through the planks into the edges of the plywood. This is the under-desk shelf compartment, which will fit around the plywood already attached to the underside of the desk.
Lay the desktop flat on the work surface, plywood side up. Apply glue to the tops of the 1-by-6 planks on the shelf compartment. Fit the shelf compartment box around the plywood on the underside of the desktop; it should fit snugly around the outside of the plywood.
Secure the shelf compartment to the plywood on the underside of the desktop with 1 3/4-inch wood screws driven every 5 to 6 inches around the outside of the top of the shelf compartment, into the edges of the plywood attached to the underside of the desktop.
Apply wood glue to one end of four pieces of 28-inch 2-by-6 lumber, which are the legs, as well as to the first 6 inches of one 6-inch face of each leg. Glue each leg to the outside of the shelf compartment, with the tops flush against the underside of the desktop; there should be one leg at each corner of the shelf compartment, on the 21-inch side. The face of the legs with the glue on the first 6 inches should be pressed against the side of the shelf compartment. Secure with G-clamps while the glue dries.
Drill four pilot holes, sized for the shank of 2 3/4-inch lag bolts, through each leg and into the side of the shelf compartment. Secure the legs to the shelf compartment with the lag bolts through the drilled pilot holes.
Sand the desk well with fine-grit sandpaper. Because of its close pores, birch sands smoothly and evenly, but it must be sanded thoroughly to avoid cross-grain scratches showing up if a stain is applied.
Apply the wood finish of your choice to the desk, as the instructions on the finish label detail. Birch takes stains and other finishes well.
Apply a wood sealer, or a protective varnish after your finish. Birch needs thorough protection from molds, insects and weathering, as it is not naturally resistant to such things, as other woods like cedar are.