First thing that came to mind is 3/4" pwf plywood, but your RS oak for sure if it's sitting around. Stuff will fall through expanded steel, no?
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My carry all is my most used implement.
I have one on my 1952, 12 speed Ford 8N and it has not been removed in 10+ years (aside from removal for service/repair). Inside the box are chainsaws, gasoline cans, hand tools, chains, clevis, etc., none of which is ever removed. The 8N/carryall is my quad.
I used a commercially available angle iron support with 3/4" exterior plywood for the front, floor and sides. Floor and front are 4x4 and sides are 4x2. I installed 2x2s inside the joints and across the top of the front board and the rear of the floor for strength. It's screwed together with drywall screws.
I've hauled tons and tons of rocks in it over the years, limiting the loads to around 700 lbs. Often I need to steer with the brakes or even travel backward to keep the front wheels on the ground.
In the 20+ years that I've used it, I replaced the plywood and 2X2s once.
Works great, and is not too heavy.
SDT