Why does one leg of my panel drop to 0v and the other rise to 240v when a load is connected?

It sounds like you have a loose, broken, or disconnected neutral at some point.

Double check all terminations are tight, and contact the conductor, not the insulation.

Also check that you have the correct earthing/bonding methods. Do you have an earth/ground wire back to the house?

If so, you need a separate earth and neutral bar, with each wire connected to the correct bar, and no link between them.

If not, you need an earth electrode connected to the earth bar, the incoming neutral connected to the neutral bar, and a link between the two.*

* I believe the NEC no longer allows this approach; I'm not familiar with US code. It may be grandfathered in.

The feeder neutral's toast

The wild voltage fluctuations you are seeing are due to a feeder neutral wire that cannot handle any significant current (effectively open). When this happens, applying a load to one leg causes the voltage across that leg to drop and the voltage across the other leg (an unloaded leg in your case, which explains the extreme magnitude of shift you see) to rise, due to the loads on the two legs forming a voltage divider.

In your situation, I would undo and redo the connections at each end of the neutral with the house breaker for this feeder off, torquing the connections screws properly at each end. If that does not correct the issue, then you will need to re-lay the cable, or use a small dry-type distribution transformer at the barn (something in the 5kVA range would be plenty adequate) to provide a new neutral point for this feeder.