I'm going to either buy or build a small barn, about 12x16, and I was wondering if it would be better to get steel or wood. The steel barns are much cheaper but I was afraid that they might heat up like an oven.
if you insulate it correctly it will not be as hot, also putting a vent in the roof helps and if you design it so that a breeze can blow through the building from any direction you should be able to get by with a steel building.
Check into spray foam insulation. An inch of the yellow spray foam will make it a lot more comfortable. There may be an insulation installer with a foam machine in your area that can do it.
i insulated mine on the south side. you could also make your overhangs wider on the sun side or the storm front side and slope it/raise it to keep it dry and maybe pour a small cement wash rack pad with a drain that is buried and is far away. you could also make gutters and drain those away to keep the mud down so say your vet or yourself could drive near it when wet. the overhang could hold your chute if you have one. i did all the above except i'm finishing the overhang for the chute.