I have a barn that is 300 feet from my house & I want to install 4 or 5 lights plus a power outlet.
What size wire & breaker would I need to do this in a satisfactory / safe manner.

Many Thanks

TKrussel will likely give you the best answer so I'll sit back and wait. My advice would be to install a sub panel in the barn and run separate lighting and outlet circuits from the panel. What do you plan to plug into the outlet, anything like a heater or compressor?

#2 aluminum will allow 40 amps for just over the recommended max of 5% volts dropped.

#4/0 will give 100 amps for just under.

Price the two and compare. For 120 volts only ,ask for 3 wire URD cable, for 120/240 volts ask for 4 wire.

You can bury it in sand 6 " above and below, for extra protection you can use 4" DB PVC conduit.

If not sure about grounding, ask.