There is plywood support that spans left to right. It sits on the lower left and right stretchers. I have a support that spans front to back in the middle. The cabinet sits on that and has a couple screws to keep it from sliding around.
The carcass is all 3/4 plywood with a 1/2″ back made with dadoes and rabbets. The drawers are all 1/2 birch plywood with 1/4″ MDF bottoms that are joined with locking rabbets (my favorite drawer joint). Full extension sildes are used.
The photo belows shows most of the stuff that goes into the drawers.



The top also acts a shelf for some of the commonly used bench tools (hammer, wonder dogs, scrap drilling board, shooting board).