Power go out when I turn key to start it won't start......

No. it wouldn't keep it from starting. If the engine cranks but won't start, probably gas or spark (ignition) related. Do you hear the fuel pump in the tank turn on for a couple of seconds, when turning key to on, not start? If you cannot hear the pump, better check if you are getting gas up to the engine. Maybe the fuel pump quit, or the pump circuit needs looked at. The pump should briefly turn on with key on to pressurize the fuel system-the fuel line and the fuel rail where the injectors are. Then the pump will turn off, and only come back on again when the engine is cranking, and stay on when running.
Correct fuel pressure is important for easy starting, and good engine performance. Have the fuel pressure checked against the specification for your vehicle. If fuel pressure is good, check the ignition system-check for spark.

What make of car?

Yes, it should. If it is conventional (like other GM cars), the pump will turn on for a couple of seconds when key is turned to on. This will prime the system for starting. When key is turned to start (cranking), the pump is turned on again at the relay, and pump stays on as long as cranking the engine happens. Once engine is started, the oil pressure switch keeps the pump running.
The best way to check if pump is good is to use a fuel pressure gauge. If it does not have a test port called a Schrader valve on the fuel rail, the pressure gauge must be teed in somewhere convenient. But if you take off the line at the filter, some gas should come out with key on, and when cranking the engine gas should come out in a steady stream.

check tank selector switch

do you here the fuel pump working when you turn the key?If not could be a fuse that runs the fuel pump.

Have you checked the automatic shut down relay ?
It powers the injectors from a 20amp fuse number 5.

I have had bad pumps rite out of the box but dont assume its a bad pump check for power at the pump with a test lite remember the pump only has power for 10 or so seconds after key is on if vehicle dont start pump will lose power until key is turned off then you have to turn key back on best if done with 2 people

yeah, try starter fluid just to verify if it;s fuel or spark related. If it starts or tries to, then it's a fuel problem. You should be able to hear your pump turn on, when you turn key on, may need to stand by tank, when someone turns key on. it should activate for about 2 seconds, with key on.(that's to pressurize your system with gas) then when cranking, the pump should run(may not hear it then-so just listen when key is initially turned on. No noise: your pump may not be working. If you can hear the pump, it may be another issue.

It could be the fuel pump. Sometimes you can hear them engage, but they are not pushing enough fuel pressure to run the car. This can be checked.

the battery indicator light usually comes on after a lose of power. Sounds to me like either bad gas, clogged fuel filter or a bad fuel pump. Easiest way to check this is have someone turn the key while you stand at rear side of vehicle and listen for the whinning noise a functioning fuel pump makes as it pressurizes the system.

P.S you dont have to start vehicle to check pump. only requires turning the key to the on position. Good Luck