The only thing you are going to find in "sheets" is plywood, fiberboard or hardboard. Neither fiberboard nor hardboard will work outdoors for more than a few days. They both absorb water and swell up and then fall apart. All 3 of them are typically sold in 4'x8' sheets so 4' squares would be the easiest to get.
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Places like Home Depot sell some smaller (2'x3' or 2'x4') pieces that look like butcher block but are made of pine that you could do something with I guess but I'm not sure what sort of glue they use on them so I don't know if they'll hold up outdoors. You could always ask them though!
Any other "real wood" would come in boards instead of sheets and would be fairly pricey to boot.