I can’t think of anything sorry
Do you have a garage? I think that’s a great place to store hay-often cooler than the house and dry.

If not, maybe a big storage bin somewhere out of the way? Closet, basement etc.

The only thing I can think of is to split the bale of hay and keep half in a wardrobe, pantry, or garage, like Kokaneeandkahlua suggested, and the other half next to the rabbits.

Can you mount a shelf on the wall next to the rabbit pen? The only solution other than boxes, cupboards or containers would be to put the hay on some type of a shelving unit on the wall. Of course, it depends how heavy it all is.

I actually read about hay storage when I was looking for a similar solution, and while I do know that hay can get mouldy (presumably from exposure to moisture), I think it is better to keep it in a dry place.

I once kept my hay in a canvas and metal laundry basket with wheels, because it’s best to have it aerated underneath, and the hay is supposed to be able to breathe in a cool, dry and dark environment. You could always put the hay on a pallet, and maybe protect it with tarp, but I would be worried that the rain, humidity (if you live somewhere humid) or rodents would ruin the hay. I once found a flattened vole in my box of hay, so I know they get into it in the fields.

I now keep my hay in the box it came, and I buy smaller quantities so it doesn’t go bad, and because I don’t have the space or strength to move a larger amount of hay, but that is just my preference. Hay properly stored can keep for many months, but I can’t remember exactly how long – maybe 6 months??