We put our big round bales up on old tires because of using the "bale bed" truck to pick them up. Many use pallets but they could get broken up and then there's a flat tire on the truck etc.. We stack them in a pyramid shape; 3-2-1 on top. The tarp goes over and DO NOT have the tarp go all the way to the ground so it "seals in the moisture". Ours come down to about 2-3 feet off the ground. We have the big tarps designed for "tarping round bales of hay". If the hay is up off the ground, it might get a little wet or some spoilage, but it can breathe.
To preserve the shape of the bales, so they don't "squish flat", you can set them on "edge" so the flat side is down. Most that store them in barns do it that way. You can put one flat side, them put one with the round side on top... like a "mushroom cap" to get the rounded top for the tarp and it won't collect water like if it was a flat top. If you have a "grab" to get them it is safer. Looks like a pair of arms. The arms on the "bale bed" of the truck works good. My son can grab them, move up a bit, set them down so they flip over to the round side down, then grab them from the other direction, and be able to unroll them or whatever. You want to round side "up" when they are outside so it sheds the rain. The thing is to make sure the tarp goes over far enough that the water will run off. The tarps that are made for that, have "loops" like large handles, on the sides that you run a piece of pipe in from one end to another or a couple of them that are 8 or more feet long, then attach something like cement blocks to the pipe and it holds it down. We tie the heavy plastic stings that come off the big square bales, to old tires, and have them hang down on the tarps on both sides; or tie tires to the pipes as weights.