Lay a 22-inch plank of 1-by-8 lumber flat on your work table. Apply wood glue to one 1-inch edge of the lumber along the 22-inch side. Abut a second piece of 22-inch 1-by-8 lumber against the glued side of the first, to create a single piece, 16-by-22 inches.
Clamp the planks with F-clamps, then lay two scrap pieces of 1-by-2 lumber on top of the glued pieces of lumber to keep them flat, and clamp the scraps down with G-clamps. Repeat so you have two pieces of 16-by-22 inch lumber.
Lay one of the 16-by-22 inch glued planks flat on your work surface. Glue a plank of 16-inch 1-by-6 lumber up on each 16-inch side of the flat plank, to make a square “U” shape. Glue the other 16-by-22 plank on top of the 1-by-6 planks, to create a rectangular box.
Drill four evenly spaced pilot holes through the top of the 16-by-22 inch plank into the edges of the two 1-by-6 planks, then drive 1-3/4 inch screws through the pilot holes. Flip the box over and screw together on the reverse side.
Fill all the screw holes with wood putty and let dry before sanding the side table box, as well as a plank of 20-inch 2-by-6 lumber, which is the back panel of the side table box, as well as the cleat for attaching the side table to the wall.
Finish the side table and back panel with your chosen wood finish and let dry before installing.
Draw a level line on the wall, an inch lower than the height you want the side table to sit at. Use a stud finder to mark out the stud positions on the line. Choose the two adjacent studs, which will be 16 inches apart, closest to the position you want to install the side table.
Sit the back panel under the wall line, over the two stud positions. Secure the back panel to the wall with three 4-inch screws through each stud point.
Fit the side table box over the back panel attached to the wall. The back of the box should go back to be flush with the wall, hiding the back panel completely.
Secure the side table box to the back panel with three 1 3/4-inch screws through each side, and five through the top and bottom of the side table into the edges of the back panel.
Touch up the screw holes with wood putty and wood finish to complete the floating side table.