How long a feeder

You need underground wire. The insulation and jacket are specially designed to withstand punctures from stones, etc. Go to your local electrical supply house, they should have it. Get the thickest gauge wire you can afford. Even with the long run, 8/2 with ground should be more than enough. You're only drawing about 3 amps, but you have to think about future use also.

Here is a voltage drop calculator

A electrical wholesale house will either have or can order 1000 foot spools of UF cable. Not likely that they will cut it.

You could always run conduit.

What you are looking for is called "UF" underground feeder cable, made for direct burial in the ground. Currently all copper wire is very expensive so unless you plan to increase the wattage out there in the future, I wouldn't go overboard on the size. 14/2 or 12/2 will be perfectly adequate and standardly comes on 250 foot and 1000 foot reels. Some electrical supply houses will cut it to order, but if not, you can buy underground wirenuts (King one step) or other brands and connect two pieces of cable

I'd run #12 wire for direct burial. Find out how far down it should go. IIRC, 12" is minimum. I'd also put a receptacle at the end. You never know when you may want to plug in an electric tool down there. You know to use a ground fault also.

I had a similar situation as the original poster. I needed a light on a pole, and decided to put a receptacle in case I needed it. I used # 8 stranded in conduit. At my supply house, it turned out to be cheaper than buying a 1000 ft. spool of #12 UF.

#10 will provide 4 amps (480 W) at 300 feet. That is sufficient to stay within the 3% voltage drop that is the standard. Use underground wire (romex or equivalent). Wire larger than #10 will start getting costly.

You could get by with #12, bulbs may be a bit dimmer than with #10, but any tool you use should be under 3 A. And what to you use for a breaker? Do they make breakers for less than 5 amps?

Why would you need a 5A breaker? Reality is, it will work, has worked, I've done it. we build a house with #12 run over 300 feet on a temporary line and used plenty of power tools.

you need to know what size house breaker, length and gauge of wire from the house, and subpanel main breaker are involved first, less the well pump power in use, to see what's available for use at the well house. then you'll want to trench and bury perhaps in pvc or conduit the proper gauge wire to your locally required depth for a least two circuits and want additional shielded cable for the camera and some conductors for doorbell/telephone/gate sensor and control circuits. you may even want to trench below the frostline and send some water there or sprinkler system lines for the future. this is a good time to check for your underground buried cables with a local hotline and make sure your gate is inside the property line on your survey. whether you'll need yellow buglites or maybe use a low voltage, decorative, LED lighting system will be options. maybe pick out an attractive distinctive fixture with an address plate and name. this project will be more attractive then the cheap dim solar units with the now dead batteries from 3 years ago.

Don't be to concerned about the slight voltage drop. This is not an inductive load. If you loose a few volts the worst that will happen is your bulbs will last longer

Sorry, I temporarily lost my mine about the breaker. On the use of tools, you can. I run my compressor and a circular saw on a long extension cord and it seems to work. But it is a lousy practice to run motorized tools where the voltage drop is more than 3%. I'm not sure what the voltage drop would be on a 300 foot run of #12 with a 10 A motor but it would be down in the motor damaging range.

I swear this thing hates me. It is suppose to read "lost my mind." If I lost my "mine," I don't remember it, so it's no big loss.