I tried formatting the memory card and even borrowing a new one but my Vita is still throwing the same error:
An error has occured. The PS Vita system will be powered off. [******]
I think this is endemic to OLED Vitas(phat ones) and it's quite sad no one seems to have any clue what causes it or how it can be fixed. I have never ever read of a Vita dying completely so I'm at a loss on why this is happening. It's a shame because it frankly renders the Vita frustratingly borderline unusable.
Before doing any of the suggestions, try removing the memory card and using it, if it still crashes, then just format the system, if it still crashes, you have a problem. If it doesn't crash without memory card, then is the memory card and not the system, you may need to do one of two things.