We store all our hay (about 35 bales at a time, usually lasts about a month or so - for the goats) in the polytunnel (smallest commercial size, so it only takes up one corner/side of it), but I wouldn't cover it with anything. I also always open the doors, front and back every day. No issues with dampness at all, and since cat is usually spending a large proportion of his day spread out on top of the bales we have found the rats have scarpered as well....
We had terrible problems with rats moving into large hay/straw bales as soon as you started to unwrap them..., but it seems not so much an issue with small square bales.