So if you light the pilot light on the water heater

[undeterred]

and it still doesn't stay lit... what do you do? Besides repeat steps 2 through 8...

http://homerepair.about.com/od/heatingcoolingrepair/ss/pilot_light_4.htm

[lizerdbits]

1. I got a new HWH since mine was 14 years old and didn't want to be fixed

Mine would stay lit to heat the water in it which by then was cold but when the flame would get lower it would cut off. How old is yours?

This says it might be a thermocouple problem. I think that's what they tried to fix on mine several times but due to its age I just got a new one.
http://homerepair.about.com/od/heatingcoolingrepair/ss/pilot_light_5.htm

[undeterred]

2. This is less than four years old. Still looks brand new.

The pilot light went out. I thought it would be easy to relight. Which it was. But it didn't stay lit.

[lizerdbits]

4. I'd have someone look at it

It's not old enough to replace.

[Orsino]

3. You put in a service call.

The symptom you describe once led to my furnace being condemned. Rust had claimed the combustion chamber.

[petronius]

5. Call the gas company - a few places I've lived, they will come out for free

to inspect appliances and light pilots (and they don't even laugh at you and refuse tips, IME). Apparently, gas companies have found that huge explosions involving their product are bad for business, so they like to make sure stuff is working properly...