Pilot light will lite. It just won't stay that way. Flame is blue, the main burner will come on. I haven't timed it to figure out how long it will stay on, but come back next morning and pilot light will be out.

My first thought is that your thermocouple is bad. It is an easy replacement if you're handy at all.

Check out my answer (https://www.askmehelpdesk.com/appliances/state-select-residential-gas-water-heater-615727.html) to a similar problem.

Best of luck,
WallyH

Hki TXwatts

Is the combustion chamber to your water heater open (can remove door and reach into the chamber) or closed (can remove door and only look at the pilot light through a little glass window)? Let me know as the answer to your issue will differ with each type of combustion chamber, OK?

Mark