In my misspent youth, I had to frequently replace the thermocouple in a furnace. I became quite adept at it, an easy enough job.
As I became older and more cautious, I called a HVAC guy on Valentine’s Day when the low that night was to be in to 20’s.
He had other plans for that night, and assured me that a pilot that lights but won’t stay is most surely the thermocouple, directed me to buy the longest one available, 36" as I recall, and install it myself. Just like old times, I had the burner in my lap and golly, it worked! 
It isn’t that hard if one of you has some mechanical aptitude, and you’ve only wasted seven dollars if that ain’t it. OTOH, I don’t want your survivors to post on your account that you died in a gas explosion.
I’m going to assume that this isn’t a fairly new gas unit, in which case servicing it’s a whole 'nuther matter. FVIR (flammable vapor ignition resistant) water heaters are now required by building codes in many places. They differ from the old style heaters in that a piezo ignition unit is provided, the access door seals shut with two screws, a special thermocouple is required, and the air intake is located on the bottom, and is a fine mesh screen.
The safety feature is that if a flammable vapor below LEL is introduced to the pilot, the flame will not stay “tight” and will wander, tripping the safety thermocouple before the point of KFB is reached. The same thing will occur if the screen on the bottom becomes blocked with lint, pet hair, or other crud, as reduced air flow creates a rich burning condition. To ensure that no enterprising soul attempts to substitute the wrong style thermocouple, FVIR units have a left-hand female thread fitting at the main gas valve.
Hopefully you don’t have one of these, as they’re more of a PITA to work on than the old standard ones.