Does the pilot light always stay lit on this water

Make sure your gas is not shut off.
Do you have any other gas appliances that are working?

If you need further help, I’m available over the phone at https://www.6ya.com/expert/gregg_27a4d257e54e7920

Under the gas valve is a removable panel. On models of that era there should also be a second panel behind the first. remove both.

the pilot light should be clearly visible.

If no pilot, turn the gas valve to OFF. wait 60 seconds. locate the tubes from the gas valve to the pilot location. turn the valve to pilot, and with a long match or grill lighter lit, place flame at the pilot location and press down on the pilot button beside the main valve.

pilot should light. If not shut off the gas and call a technician. If yes, hold the pilot button for 60 seconds, then slowly release. Pilot should stay lit. If not, shut off gas and replace thermocouple, start relight sequence over. If pilot stays lit, replace inner cover and adjust temp to 120 deg. If heater fires up, replace outer cover. Done.

If heater does not light, replace gas valve or entire unit. Water heaters have a limited life expectancy, and if my recollection is correct your unit is 10yrs past warranty.

Make sure thermocouple tip is seated inside the burner flame.
Make sure thermocouple is not overtightened. Finger-tight + 1/4 turn.
If the pilot lights, then ECO energy cut off is not tripped, but gas control valve might be bad.
Check color of pilot flame: it should be strong and clear blue.
If pilot flame is weak, or color of flame is yellow-orange: Clean pilot orifice and burner and combustion parts on yearly basis.
Clean air intake screen if applicable.
Increase air supply to water heater. Open a window and see what happens.
Check for adequate draft going up chimney using match under vent hood to see if smoke drafts immediately upwards.
Bleed air out of gas line.
Check gas pressure.

Open following links for troubleshoot resources:
http://waterheatertimer.org/pdf/Pilot-light-will-not-stay-lit.pdf
http://waterheatertimer.org/pdf/Pilot-outage-troubleshooting.pdf
http://waterheatertimer.org/pdf/Vent-spillage-test-atmospheric-gas-water-heater.pdf
http://waterheatertimer.org/How-to-install-gas-water-heater.html#troubleshoot

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?

If I understand correctly, the pilot light is not staying lit.
You light the pilot and water in the tank reaches temperature.
Once hot water is gone, then pilot light has to be lit again.

Gas pressure is good, and gas is flowing.
Thermostat is working at high end because it cuts off heating.
Vent is working because tank heats up.
Air intake is working.

Gas control thermostat might not be reading low temperature. Replace gas control valve.
Pilot tube might be dirty. Clean burner parts and combustion chamber, and air intake, and vent.
Backdraft might be blowing out pilot. Check for constant updraft by running water heater 5 minutes, and then light match under vent hood and see if smoke goes up.
Very hot air can put out pilot, especially when water heater located in attic. Add ventilation.
Improper vent might have water dripping back down into combustion chamber and putting out pilot light and rusting out the parts. Inspect for rusted parts and water.
http://waterheatertimer.org/How-to-install-gas-water-heater.html#troubleshoot
http://waterheatertimer.org/pdf/Pilot-light-will-not-stay-lit.pdf

More than likely the thermal couple is defective. The thermal couple is a safety device that shuts the gas off if the pilot goes out. If it is defective the pilot light will not stay lit. The thermal couple is the device that the small copper line goes to. It is mounted beside the pilot flame and the flame much touch the TC for it to work. If the flame is not touching the TC bend it until it is in the flame. If it is touching the TC then replace it.

The gas valve makes a single click sound when the pilostat shuts off. That is a safety holding coil for units with standing pilot lights and closes if it thinks the pilot went out. In electronic ignition, the same holds true except the sensor is electronic.

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.

It sounds like your main valve. The diaphram may be loose fitting or it has a small leak.

First Clean the ignitor assembly and make sure the flame is reaching the flame sensor strongly.

You may need to replace the flame sensor.