It sounds like the shop is talking about the fuel pump. As others have stated the fuel level sensors will not keep the car from starting or running.
Assuming it's a fuel delivery problem, then it's probably a known problem with 370Zs - with the tank level at ~1/4 or less, on a hard right turn the fuel moves away from the pump inlet causing starvation. No need to replace the pump, just keep the level above 1/4.
If you were driving "normally" when it happened, you may have a bad fuel pump.
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