Different storage methods were used and testing indicated the level of feed quality losses using each. Here's how various storage methods of large round bales fared in the UT research:
Stored on the ground with no cover, 37% loss.
Stored on tires with no cover, 29% loss.
Stored on the ground and covered, 29% loss.
Stored on tires and covered, 8% loss.
Net wrapped and on the ground, 19% loss.
Stored in the barn, 6% loss.
The research shows that losses can be significant depending on the type of storage used. The data also suggest that storage losses occur from moisture getting into both the top, and bottom of the bales. Thus, if hay can't be stored inside a barn, the next best option is to at least get the hay off the ground and under a tarp or plastic cover. Raymond L. Huhnke, Oklahoma State University Extension agricultural engineer provides more information on large round bale storage here.