That's a lot of under ground cat 5 ethernet wire to run, and would that be very expensive.
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cat5" is a very old standard which you actually don't want anymore. "cat5e" is the modern minimum. A 1000 foot box of this cable typically costs about $100 to $200 USD and you can find it rated for outdoor and direct burial. Avoid the "CCA" or "Copper clad aluminum" variety even though it can be cheaper. Solid copper is better.
By the way how deep would you have to bury the cable.
"PoE" (power over ethernet) is low voltage and isn't going to hurt anyone if it is accidentally cut, so I think nobody really cares how deep you bury it (or if you even bury it at all). You get to decide if you're going to bury it, how deep, and what kind of conduit you want to use (if any).
When I ran cat5e out to some planters in my front yard, I just tucked it out of sight under the coarse gravel landscaping rocks and called it a day. Didn't even need a shovel. Of course I wouldn't want to do that for a 300 foot span.
By the way i do have power at the barn, i was just looking for the easiest and the cheapest way.
Since you have power at the barn, forget about using wifi cameras or running a long cable. Get PoE cameras instead, and link them back to the house using a directional radio bridge like
@TonyR suggested. Do note that both suggested radio options use 24 volt passive PoE. This is NOT COMPATIBLE with standard 802.3af PoE than cameras use. You need special 24v passive PoE injectors for those. The TP-Link radios come with suitable injectors, but the Ubiquiti 5AC radios do not. So if I were you I would probably just buy a pair of the TP-Link 5 GHz kit for $50 per radio ($100 total). You will also need several cat5e cables. Two for each bridge radio (one into PoE injector, one out to radio), one for each camera. Best to cut and crimp the cable yourself if you don't like drilling big holes for prefabricated cables.
Here are a bunch of products to consider.
TP-Link 5 GHz bridge radio (you'd need two of these):
Cheap solid copper cat5e cable (1000ft):
RJ45 connectors,
crimping tools:
Cheap PoE switch for cameras:
My favorite overview camera:
ANNKE 6MP POE Security Camera Dual Lens Outdoor Color Night Two-Way Audio FCD600. Condition is New. Shipped with Standard Shipping.
www.ebay.com