Why do I see 240v between hot and neutral when one breaker is out in my mobile home?
Neutral to ground should not be 120.  It should be at (or very near) 0.  Neutral and ground are generally bonded at the service panel.  When only one breaker being on causes this (and the other doesn't), it seems unlikely that you have lost your neutral service from the supplier.  There is likely a fault on the neutral associated with one of the circuits.  It's also possible that there is a faulty or mis-wired MWBC (explanation of MWBC here).

You could try disconnecting the wires from the breakers in the service panel and testing directly from the breaker to the neutral and ground.  If everything tests as expected, you're looking at a wire fault.  Otherwise, the wire isn't the first problem to fix.