I am the 3rd owner of the car and it’s got about 62,000 miles on it. For some reason when I try to start the car sometimes the battery just dies. There’s no power to anything, and I can’t even use the powerlocks. Just now I had the door halfway open and pressed the clutch into the floor and turned the key and the power just went dead. Is this just me, or is it a common issue? What am I doing wrong?

 

 

Could be a wiring fault, but could also be the security immobilizer. I suspect wiring fault since the immobilizer shouldn't kill power to everything. Start with the simple things first.
1) Get the battery load tested.
2) Load test the charging system while the engine is running.

If either of those fail, that's most likely the issue.
Else, if both check out good, then you may have a short somewhere. Trace the connections between the battery and starter. Check the main grounds in the engine bay.

To me this is a very unusual problem. Electrically speaking, 350Zs are very reliable (excluding environment such as rust belt areas)...however if you're the 3rd owner and previous owners installed aftermarket alarm systems, stereo, or other stuff, there's a good chance you have wiring hack job going on behind the dash or in the engine bay that may need sorting out.
-Icer