Sorry I'm a bit late to the party.
Here is a solution that worked for me some years ago. It is cheap, and requires minimal carpentry skills (I've moved on a bit since). The drawer bottoms have held up well with minimal sagging. The system provided quick and easy access to the drawer contents.
Here is the frame. The upright sheets are 15mm chipboard. The frame is made from 2×1 sawn, so no precision there!

Here are the drawers. The drawers are made from 2×1 sawn, which are joined together by butt joints, a screw, and some glue. The drawers are around 500mm square. The drawer bottoms are 6mm mdf, screwed to the drawer sides. The drawers (trays?) just slide on 2×1 sawn.


Here it is in use.

Steve
Edited By Steven Vine on 11/07/2017 12:54:08