There's no reason why you can't build cabinet drawers; they're nothing more than boxes. The only differentiating factor from a typical box is a dado or channel at the bottom on both sides. The dado supports the drawer bottom and the drawer's contents. Use simple equations to determine the size of the drawer, and build it using common materials. You don't need fancy joints. Contemporary cabinetmakers know that complicated joints aren't necessary on drawers; they're typically for aesthetics. The simple butt-joint combined with modern glue and fasteners provide all the integrity you'll ever need.
Measure the height of the opening and subtract 1 inch for the height of the drawer sides. Measure the depth of the opening from the inside to the back of the cabinet and subtract 3 inches to determine the depth or front-to-back dimension of the drawer. Measure the width of the drawer opening and subtract 1½ inches to determine the width of the drawer -- the length of the rear panel.
Rip four pieces of 3/4-inch-thick hardwood plywood to the height of the drawer with a table saw. For example, if the drawer opening measures 6 inches, rip the plywood to 5 inches. Cut the two pieces to length using a miter saw. If the opening depth is 22 inches, cut the two pieces at 19 inches. These are the sides of the drawer.
Subtract 1½ inches from the width of the drawer to allow for the thickness of the side pieces. Cut the remaining two pieces of plywood at the measurement with the miter saw. If the drawer is 14 inches wide, cut the pieces at 12½ inches. Subtract 3/4 inch from the width of the two pieces. Rip them to the measurement with the table saw. If the two pieces are 5 inches wide, rip them to 4¼ inches. These two pieces are the front and back of the drawer.
Install a 3/8-inch dado blade on a table saw. Raise the blade to 3/8 inch from the top of the table to the tallest tooth on the blade. Lock the fence down 3/8 inch from the right side of the blade.
Run the two side pieces -- the long pieces -- over the blade to create a dado or channel 3/8 inch from the bottom, 3/8 inch deep. Stand them on edge, parallel to each other with the dado at the top, facing in.
Apply wood glue to the ends of the front and back pieces. Place them between the side pieces flush at the ends. Shoot four 1 1/4-inch staples, evenly spaced, through each side piece to penetrate into the ends of the front and back piece.
Measure the distance from the inside of the dadoes from side to side. Measure the length of the drawer box from front to back on the outside. Cut a piece of 3/8-inch hardboard to the measurement with a table saw. This is the bottom of the drawer.
Apply a small bead of hot-melt glue to the inside of the dado channels on both sides. Slide the hardboard into the channels from one end. Flush it on the ends. It will slide over the front and back piece. Shoot six 1-inch staples, evenly spaced, through the hardboard to connect it to the front and back pieces. The drawer box is complete.
Measure the drawer opening and add 1½ inches to the width and height. This is for the drawer front; its dimensions will be 3/4-inch bigger around the perimeter than the box's dimensions. Cut a piece of hardwood lumber to the measurement. Sand it smooth using 100-grit sandpaper. Round and blunt all the edges and corners.
Drill four holes through the front of the drawer box, evenly spaced at the top and bottom. Use a drill/driver and 3/16-inch bit. Insert 1¼-inch screws in the holes. Center the hardwood drawer front on the drawer box. Drive in the screws to secure the drawer front to the drawer box. Sand the box smooth, round and blunt all the corners and edges.
Place your choice of drawer pull hardware on the front of the drawer. Trace around the contact points. Drill through the drawer using the 3/16-inch bit. Screw the hardware on with the screws provided with the drawer pull.
Adjust the length and width of the plywood, hardboard and lumber according to the dimensions of the drawer's cabinet or chest opening.
The drawer front is an example of a full-overlay drawer front. Cut it smaller for an inset drawer front.
Drawer guides are optional. For this type of drawer, bottom-mount guides work best.
Wear breathing and eye protection when working with wood.