You can run Cat6, even in the same conduit if you want, but STP doesn't solve your major problem:
Ground loops.
You'll need protector blocks on both ends suitable for Cat 6, as well as solid state secondary protectors to deal with sneak currents and the like, which you will absolutely inductively couple into the cable from the AC power.
While technically doable, running copper (safely and correctly) between two buildings, is almost never worth it. If you're going to do this, I'd suggest running (and having a professional terminate) multimode fiber, or use wireless.
Fiber's a nice way to go, it's EMI/RFI immune, doesn't need protector blocks on both ends (which are getting rare and expensive these days), fast, and not really any more expensive than good outside plant cable.
Also, realize if you're running power out to a shed, you'll need to pull permits, have it inspected, and you'll owe some application fees.
I certainly don't recommend it, but I've seen plenty of people (even contractors that should know better) do it. Shoot, I've even seen one try to run Cat5 600' joint buried with 200A service. Wondered why the phones weren't real happy.
