DynaGlo Heater S05-010271

If the pilot light won't stay lit and the thermocouple is okay, then the thermostatic control valve may be defective. Replacement of this part should be performed by a trained gas technician. ... If your gas water heater doesn't stay lit, you may have a defective thermocouple.
https://www.fix.com/repair/appliance/water-heater/wont-stay-lit-keeps-shutting-off/

Sounds like the thermocouple is shot

1) Get more air to the water heater and see if problem continues.
2) Read the following pdf-s for things that might apply to your situation.
http://waterheatertimer.org/images/Negative-pressure.jpg
http://waterheatertimer.org/pdf/Attic-Ventilation-and-Pilot-Outage-in-Gas%20Water-Heaters.pdf
http://waterheatertimer.org/pdf/Pilot-light-will-not-stay-lit.pdf

3) Read pdfs above first, and then Add a comment with missing information below, for possible do-it-yourself help.
Does this water heater have a blower?
Is gas valve electronic? Or does gas valve have a dial?
If gas valve is electronic, have you checked outlet with circuit analyzer?
Did pilot light work correctly for a while? If so how long?
Where is water heater located? Hot attic? Closet? Garage? Is area dirty and dusty? Has air intake been cleaned?
What color is pilot flame? Clear blue or yellow-or-orange?
Have you inspected inside of combustion chamber for dripping water?

thermocouple probablly bad, is the flame covering the thermocouple.
the flame has to hit top portion of thermocouple to work correctlly.

Try replacing the thermo coupler.There should be a little bulb shaped part(like a pencil eraser) that the pilot light heats up .The other end screw into the gas valve.If this is going bad or has gone bad the pilot light will not stay lit.Most of these are under$10.Well good luck and try to stay warm.

Probably a bad solenoid in the gas valve. Higher voltage of a new thermocouple keeps it open longer but it still fails over a period of time. Look into a new gas valve for the water heater.

YOUR WATER HEATER HAS WHAT IS CALLED A STANDING PILOT-STAYS LIT ALWAYS WHEN THE PILOT GOES OUT YOU LOOSE EVERYTHING. YOU HAVE TO REPLACE THE THERMOCOUPLE WHICH KEEP THE PILOT LIT. ONCE THE PILOT IS LIT THE THERMOCOUPLE SENDS IN MILLIVOLTS TO THE BURNER TO SAY ITS READY FOR BURNING
IF YOU ARE NOT FAMILURE WITH THIS REPLACEMENT PLEASE FIND SOME ONE WHO CAN. IT IS NOT HARD TO DO ASK A NABOR OR FRIENDS SOME HAS TO KNOW HOW. SERVICE COMPANYS ARE OUT FOR THE MONEY FOR A 10 MINUTE JOB

Cleaned pilot light area with an air compressor. Problem solved!

crappy pilot flame (not properly heating the thermocouple), bad thermocouple, thermocouple loose where it screws into the pilot safety valve, bad safety valve magnet.

One or more of those is the problem.

Oh...there is also a saftey switch...in case it tips over. It will also cause the pilot to not stay lit (if the switch or wires that go to i are faulty)