I have a couple of different things to try this year. According to the book The Lost Country Life, here is how an English haystack was built:
A stack of hay always had some foundation; it could be built upon a level flooring of stones, covered with a deep layer of still-green bracken, which does not seem to have 'risen damp' to the hay... The reasons for this bracken foundation were that through it the air could pass easily (it does not pack close like hay), and that it raised the valuable hay several feet off the damp ground and was rat-proof -- rats will not gnaw through bracken, for it, like horse tail, makes their mouths sore.
I don't know that we have actual bracken around here, but there are several different kinds of ferns. The hillside across the road has quite a few, and I have permission to cut as many as I want. It's very labor intensive to gather the rocks for the foundation, so I'm only building one of these haystacks this year. I calculate that it will hold approximately 1000 lbs of hay, which is 1/5 the amount I need for the goats.
I have also sketched out some ideas for racks that would store hay completely off the ground. I'll describe it more fully and take pictures once I've built them and made sure that they work, but basically there would be a frame to hold my large tarp off of the top of hay, and several smaller frames underneath to hold mounds of hay off of the ground.
Finally, I plan to get large burlap bags and pack some of the hay into them for storage in our garage. If I get the construction rolls (6' X 1500') and cut them down into 3' X 6' bags, each bag would hold approximately 50 lbs of hay, or comparable to a bale. The bags would be for backup, in case anything happens to the hay stored at the field. I would probably also use them on days with heavy snow or ice, where I would be unable or unwilling to get the hay out of the haystacks. Buying the burlap for the bags will be slightly more expensive than just buying that quantity of hay, but the bags would be reusable for many years.