Its why you test before you trench. Cat5e is supposed to be good for a 100m run of solid core cable with a 10m stranded Cat5e cable on each end and stand up to full gigabit speeds. If you are running solid core from switch to switch (or router, or any other active device) you should in theory be able to do at least 120m direct connect. Possibly even a bit longer with the right NIC and the right run. I certainly wouldn't chance more than 100m, but in theory it might work.
90m is the max not including the patch cables. Below is a quick quote for wiki. I can't speed a lot of time reviewing my network cabling course I took ages ago
Under TIA/EIA-568-B, maximum allowable horizontal cable distance is 90m of installed cabling
With Cat6, the wire gauge is generally a step higher and it is also generally less susceptible to interference. Its also only rated to the same 100m+2x10m for gigabit as Cat5e, but I'd bet my bottom dollar it can do more than that. On simple impedence it should be able to manage at least 20% longer due to the larger wire gauge.
The 20% you speak of is not code
All of the suggested installation techniques are nice, but they are also in general massive overkill, especially for a single wire installation. I've tested a lot of less than ideal setups, like running a wire for 50ft right on a 240v line and zero detectable difference in performance versus having it located 3ft away, or against a 10ft run with no electrical lines anywhere near it. Now, that might change if it was a 300ft run, or if there was something VERY noisy on the 240v electrical, but I've tried hard to induce interference and nothing has batted an eye on gigabit over Cat5e.
Typically power is in a grounded conduit but running with power unshielded is not code
Cat6 and Cat6a would be even more immune to such things. Granted, I am not running a server room with hundreds of cables or trunks with 20 cables bundled together, but at least in a home setup, its awfully hard to induce interference that has a measurable impact. That doesn't mean you shouldn't follow best practices where it makes sense to.
For shielded, you need to be very, very careful running shielded below ground as THIS can setup a voltage differential in your home's ground setup depending on how you are grounding it out at the end points.
Honestly based on cost, I'd just get a couple of transcievers, a couple of mini GBIC fiber adapters and run fiber for the length and then you don't need to worry about any of it.
~300ft though is well within the tolerance of Cat5e to carry gigabit at full speed (its 328ft+ a pair of 33ft stranded cables on either end).