Thanks for the plug!
Don't tarp good hay! It just doesn't ever work except when you have really dry hay and a really good tarp - not a blue walmart special. And generally it only works well in the upper midwest or say the southwest where the climate is already arid.
Building a good shed is a good way to go.
Buying Portable sheds or portable carports with actual roofs on them works ok.
Used semi trailers work well and can be had for $1000-1500. If you leave them on the wheels, the floor will stay nice and dry. Need to cut and install vents in them to make them work well, but the storage is nice and lockable. Look at cubic feet on your options real close as a 40' reefer will look nice and big but have less useable space than a 28' dry box van.