If it were me I'd use the 8/3 direct burial mentioned by Fly320s and do a 220 to a new breaker box. You would be slightly under 3% voltage loss at 60' with 10 gauge and 120, but that doesn't account for wiring in the structure or if it is really a bit over 60'. With 8 gauge and 220 you should have little voltage loss, probably less than 1%.
I did something similar for a friend to light up her riding ring at her horse farm, but longer distances. I did a 220 run about 200' feet out to the riding ring using 4 Gauge direct burial aluminum through a relay in the barn and then split it into two 120 circuits that ran another 200 feet or so for to the three sets of lights on both sides of the ring.
In the spring I'll be doing almost exactly the same thing you are doing as well running electric out to my barn, but that is about 180' for me.