Fuel Pump or timing belt. Try popping the lid off the air cleaner box, and with a can of starting fluid, spray it into the intake while someone is attempting to start it. If the engine can run off the starting fluid, change the pump and filter. Pull the pump out first because there are two different types of pumps that were offered with this car.
If it's the belt, the noise from the engine will be a bit higher pitched when attempting to start, as the crank is spinning, but the cam assembly is not. Good thing is I don't believe GM made an interference engine, so there shouldn't be any costly valve-to-piston contact.
Another, although less likely problem could be a burned up ECU or PCM. Both can be extremely costly as the keys would no longer work.