Turn the table upside down on the top of two scrap 4 by 4-inch boards so that the table is stable and elevated off the floor 4 inches. Measure to find the center of the table. Mark the size of your recessed opening. Draw the shape of the recessed box on the underside of the table.
Drill a hole inside the box large enough to fit a jigsaw blade. Cut out the box shape staying inside the line 1/16-inch on all sides. For purposes of this example, our recessed box will be 4 1/2-inches wide, 18-inches long and 3 plus-inches deep.
Cut one piece of 1- by 6-inch board 19-inches long using a table saw. Cut two pieces of 1- by 3-inch board 19-inches long.
Place the two 3-inch boards parallel to each other on the work table. Stand the boards on end so that the 3-inch side is upward. Apply carpenter's glue to the top edge of each board. Align the flat side of the 6-inch board with the outside edges of each board. Nail through the flat side of the 6-inch board into the glued end of the 3-inch board. Nail every 4 inches. Drill pilot holes between your nails and screw wood screws to hold the boards more securely. Wipe off excess glue with rags.
Turn the boards over so that the sides are pointing upward. Measure between the upright 3-inch boards along each end. Cut 3-inch boards to fit. Apply glue along 3 edges of the end board and press the board between the sides. Align the outer edge. Nail and screw the boards together on each side. Repeat for the second side. Wipe off excess glue with rags. Allow the glue to dry for 24 hours.
Turn the wood box upside down over the hole in the underside of the table. The hole should be slightly smaller than the box. Line up the sides of the box parallel to the table. Draw guide lines around the outside of the box. Lift up the box and apply glue to the top edges of the box. Press the box into place using your guidelines.
Screw two L-brackets along each long side and one L-bracket at each end to attach the box to the underside of the table. Be careful to use short screws that will not come through the table top. Turn the table right side up.
Cut a finished inside edge around the inner lip of your recessed area using a router. Allow the router to guide itself along the inside walls of the container.