We just finished off my husbands shed and used underlayment from HD. It is thin ply on both sides covering a thin foam. It is less than a quarter inch in thickness. We used it on the walls and ceiling cut in strips (12") to look like boards run horizonal and solid with a poly protective covering for the floor. A 4x8 foot piece runs $15 a sheet. You could run cut strips vertically between the framing on your barn. HD cuts it free just take in your measurements. The cuts do need to be more than 12" wide though. You'll notice one side of the underlayment is knotted like boards and the other side (used for flooring) is more solid.
Here's what the wall looks like (untreated) and the floor with poly. It worked really well for us since it is only an 8x8' shed. I also used the same underlayment in the office space (converted garage) for a wall (10x10') but painted the strips white. That wall gets a bit of moisture from old plumbing (air vent pipe shown) so I didn't want dry wall. It has held up well.
. Good luck with whatever strikes your fancy to use. It is nice having that extra little space outside the house.