I have a gas water heater that the pilot light won't stay on?

You may have a thermocoupler problem, or scale (dirt) on the jet or orifice that needs to be cleaned. But I agree that you should probably check with a professional to fix this. Gas problems can cause MAJOR issues (almost blew up the house, here, from a teeny tiny leak in the oven!).

Be careful and hope you get it fixed soon!

You more than likely need a new thermo-couple. It is the brass wire that runns from your control box to the pin that set right beside your pilot light. The heat from the pilot light heats the pin and causes a reaction which keeps the pilot lit. If the thermo-couple goes bad, the pilot won't stay lit. Go to the hardware store and tell them what kind of water heater you have and how long the brass wire is, and they can sell you a new thermo-couple to fix the problem. I have installed several over the years. It is not difficult, and the instructions are easy to follow.

The pilot when it lights may have a week flame barely covering the t-couple end. The flame should cover 1/4" of the end. adjust to do so. bend the pilot bonnet or assembly, closer to the pilot flame. When the main burner lights, it may starve the pilot becuase the pilot orifice might be blocked w/ sulfur. disassemble and clean.

If the heater has an auto-igniter, then the igniter plate may have crust (From scale, etc., that came loose when the tank cooled.) If this has fallen on the plate, the auto-igniter is shorting to the crap on the plate, and not igniting the gas. You need to clean out all the junk that came from the tank cooling. Then the auto-igniter will do it's thing and you will be back in business.

Did you replace the cover? Some time the draft extinguishes the flame.

Call a professional. You don't wanna mess with gas problems.