My Honda G300 engine runs for 5 minutes and then cuts out

Low oil.
Check oil level should be to bottom of inspection plug threads or half way up dipstick threads when unit is level, motor level. Add oil if it is low (don't add too much as it will cause damage to unit.)
Try running with oil added if it still shuts off unplug yellow wire to 13mm bolt that enters crank and try running again
If it runs for longer or runs fine you can leave off wire. As it is a safety feature, just check oil frequently to make sure it isn't burning off oil which could lead to engine seizure if unchecked.
If engine runs fine at low speed and dies once under load it might be a clogged main jet. Drain carb bowl and spray where bowl fastening bolt screws in with carb cleaner, caution do not go crazy as the gasket that seals the perimeter of the carb bowl tends to expand on some models ( so use good aim) 're attach.
Start and make sure you're choke isn't on, that also causes start, sounds good then becomes rough idle then shuts off, basically it's a butterfly valve which causes more suction through the main jet. If left shut it floods the engine open up the air filter and you should be able to see the valve and test it's function, if you've taken the carb apart before this could be your issue.
Good luck

Also you can check for the low oil safety
Shutdown.

You have 1 of 3 possible issues; a dirty carburetor, a clogged fuel line or your ignition module (attached to your plug wire) is going. If you can start the engine back fairly quickly it's probably the fuel. If it takes 10-20 minutes to restart, the ignition module.