We had a carpenter make a wooden box similar to this to store our boxes.
AWe initially tried storing cake boxes vertically but they started to warp over time, so we had a cabinet built (we used California Closets, but if you are on a budget you can probably find something similar at Ikea) that was wide enough and deep enough for the largest box (half-sheet) folded in half with several additional shelves for each size.
You can buy the closet wire shelving at Lowes or Home Depot really cheaply, I built out my under-stairs closet with it. It comes in various sizes too.
I found a cheap art display stand (I think I paid ~$40 for it on an online clearance). I can't find a pic of the exact one, but this is similar. I found the pre-fabbed shelving units weren't deep enough for the large boxes. I do stack my smaller boxes on shelves but this works for the larger ones.
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