Is there a groove/track in the bottom of the door frame? I have seen barn door hardware (some with rollers, some just a guide) meant to keep the bottom of sliders from swaying. There are zillions of different hardware doodads for sliding barn doors. Worst case, you could bolt some track on the bottom of the door and then put a guide on the ground. Could be as simple or fancy as you want.
I have a barn set to start in May. If I don't have budget for concrete, my plan is to put down plastic, then gravel (probably #57 or #78, which self compact with minimal work), and then crusher run. That should work fine for utility use, but not sure if it would be OK for horses though. May not be good to stand on, and certainly would not be easy to clean.