You don't have to replace a sagging drawer bottom. A typical drawer has a cavity beneath it that's just deep enough so you can strengthen the bottom with a piece of plywood. The following article explains this process which will save you time in repairing a sagging drawer bottom.
Make sure the drawer box is square by using a large framing square or by taking diagonal measurements corner to corner. Equal diagonal measurements means the box is square. If the box isn't square, square it up, clamp it and drive brad nails through the bottom. Two brads placed near the middle of each side usually provide enough strength to hold the box square.
Stiffen the old drawer bottom with any 1/4 inch plywood. A full 4 x 8 foot sheet costs about £6. Many home centres also sell half or quarter sheets. Measure between the drawer sides, front and back and cut the plywood 1/4 inch smaller than the opening (1/8 inch on each side).
Glue the new panel in place. If the underside of the bottom is unfinished, use wood glue. If it has a finish, you can sand the finish or use construction adhesive. Set books or other weights on the panel. Wood glue forms a strong bond in about fifteen minutes. If you use construction adhesive, leave the weights in overnight.
The above allows for an alternative to completely replacing a sagging drawer bottom. It is effective, efficient and allows one to keep the look of original wood in the inside of the drawer box. This is especially important to those who want to try to keep older furniture as original looking as possible.