Best Way to Extend 30A 120v circuit over 300 feet?
Concentric-Neutral cable is relatively inexpensive and is intended for direct-burial. Personally, I'd be getting the 2 AWG stuff but that may be more money than you want to spend. 

Note that you will need to drive ground rods at the barn. Two rods a minimum of 6 feet apart should be good. 

Be sure to both bury it deep enough and put a pressure-treated board over top of the cable before you pile the dirt back on top.