I think your falling short on the assumed * AMPS that will be used but lets assume 120/240.
50 Amps 350 kcmil XHHW 2 1/2" PVC sch 40 4-wire under 3% VD
50 Amps 4/0 CU XHHW 2" PVC Sch 40 4-wire under 3% VD
* You should probably load this up to 60 Amps. I only gave you 50A.
You need to find out all that is going into the Barn, sometimes they turn this place into
a very comfortable place, Office, Bath room, Hot-water.
Then just consider the standard stuff like Heat, A/C, even a motor for a barn fan, lights inside and out, then the real equipment like a welder, car lifts, Receptacle circuits, Etc., Etc., future
Bottom line you have to state you need to know exactly what's going in there, because your
suppose to do load calcuations that account for all the equipement, or worse future stuff.
that this install gets into.
Beside at this distance the price of wire and a transformer in barn might be a real option!
Which gets back in types of loads - equipment.
You don't want to embarass them or yourself in this matter.
Welcome to the Forum.