We've got three systems engineers over here discussing your issue on a slow day. We've argued a bit, but reached a consensus:
1. A 350 meter run is far too long for Cat5e ethernet. Not an option.
2. Cat6 would give a more reliable signal, but you're still beyond the specs. At best, you'd get an intermittent, unreliable signal. Poor option.
3. The limits of coax are typically rated at 200 meters, though some say you can push that to 300. Even so, our cable expert here says it's a no-go. I hate to agree with him, but I do.
4. The classic suggestion is to go with fibre. (Which I assume you are trying to avoid due to cost). For that distance, it wouldn't even have to be a professional fibre job. You could probably cut it yourself and get a great signal. But, you've got the "bury the cable in PVC pipe" issue, just as you would with any cable solution.
5. My suggestion would be to use a wireless "signal extender" or wireless access point (WAP) in your barn. You could run ethernet to the barn from either direction (if its under 100 meters), hardwiring the WAP. Out in the country you shouldn't have a lot of interference. You might be able to make the 100-meter plus hop, or ethernet wire out to a second WAP from the other direction too. It would cost less than fibre, a lot less installation headaches, and if it worked, it would be more reliable than an "iffy" coax solution.