I don't know about room. I've never stored that many square bales, but the main reason for storing the hay away from the horse barn is fire hazard. A barn full of hay is basically a fireball waiting to happen and all it takes is one tiny spark to start a monstrous blaze...and it doesn't take long for it to go completely up.
I thought that, too, but when I first got my horses (June, 1985) I rented turnout with a barn for shelter from a farmer. He also rented his previously run father's chicken barn buildings to lot of other businesses, and one of them started a fire in February, 1986. Afterwards, I saw my gutted pop-up camper, bits left from leather bridles that had burned up, no more wooden desk (that had my business phone), no phone either. Most of the bales of hay and straw were singed on the outside, but intact inside, to my great surprise.
You know, LOTS of things are combustible in a barn. Barn fires from hay occur mostly bc of wet hay that is improperly baled and/or left out in the rain, and then stacked. I have bought "rained on hay", and opened up the bales, turning the flakes until they were completely dry. You CAN do that. You might HAVE to do that this year bc in many places the drought has left hay scarce, so I mention this. The temperature inside a wet bale a couple days after improper curing will burn you. It is amazing how hot they can get.
I have lightning rods on the roof of my barn. The builder (70yo?) put in a loft with stairs and I can store up to 500 bales in it. I keep my horses in their stalls below the loft mostly when there is ice or a snowstorm and I give them lots of turnout year round with their 16 x 19 ft. Shelter, adjacent to the barn.
Put up dry, properly cured hay. Don't smoke or let anybody smoke in or close to your barn. Don't stack hay on a dirt barn floor bc your floor could get wet and ruin your hay with mold. Mold from properly cured hay doesn't start a fire, either, but a horse won't touch it. I suggest wooden pallets to get some air under the bales. Cattle will eat it. Most of us stack bales on their sides to keep moisture from molding next to the baling twine. Just wanted to share, so you know.