System is shutting down during OS upgrade

I think you have a deeper problem than the hard drive issues. Just to be clear here your system boots up and the screen shows a folder icon with a blinking question mark in it and if you leave the system on it stays this way (plugged in). If the system shuts down on its own you likely have a logic board failure.

If the computer stays in this condition we can try to fix the drive issues. Do you have a second Mac or a friend who has one? You need to find a Mac with a FireWire connection (the newer systems don't have it any more). With a FireWire cable you'll connect your system to the other, then when you boot up your system you press the letter T to get your system into Target Mode. That way your system is seen as if it were an external drive to the other system. Once in this state you should be able to check your current internal drive using Disk Utility and if you need to salvage any of your files you can also copy off as needed.