This is the failure people describe in two ways. The pilot will not stay lit. Or, in Spanish, el piloto se apaga al soltar la perilla. Same heater, same part, most of the time.
You can light the pilot. It stays on while your hand holds the knob. You release the knob and the flame dies. The thermocouple is the sensor that tells the gas valve the pilot is actually burning. When it is worn out, the valve closes as soon as you let go. A thermocouple or gas-control repair typically runs $175-$450 if the rest of the heater is sound.
Stop and leave the house if you smell gas. Call the gas company. Do not keep relighting a heater that will not hold a flame and do not take the burner apart.
A different problem is a tank that lights, runs, and then you get only a few minutes of hot water. That is often sediment, a failing dip tube, or a heater that is simply too small or too old. A puddle under the tank, with dry fittings above it, is a failed tank. Replacement of a standard tank in San Diego typically runs $1,400-$3,200 installed, and the heater has to be strapped.
Hard water in this county shortens heater life. If the unit is over 10 years old and the pilot is already failing, ask the plumber to price a replacement next to the repair so you can see both numbers. Many cities here, including the City of San Diego, require a permit for a water heater change-out. Confirm that on the city portal.
We match the call. The plumber does the work.