There is nothing that will save a job necessarily, if the terminating party is bullheaded enough. But email, if forwarded to an account you retain access to after being terminated can serve as powerful evidence in follow-up litigation and for that reason alone, it can save your butt in a situation where they say one thing and you have another in writing. Because while yes, printed emails can be faked, hacking Gmail's servers to inject fake emails is considerably more difficult, making any hanky-panky with the email database on the employer's end a lot more obvious.
It might not be super-hard to fake an e-mail from a given IP, but usually the timing of a critical communication can give it some veracity, especially if it agrees with what's been found on other servers, so forward anything you think is important sooner rather than later when you get the sense trouble is brewing.
For nipping problems in the bud at work before it comes to an actual risk of termination, however, any email evidence of bad behavior on the part of a coworker or good behavior on your part that refutes what's being said about you should be reported to your immediate manager or whoever is over their head if they're the source of your problem. And do it promptly, because it's sad but true, a lot of people respond more to a narrative that's been formed about you than evidence you present that counters all of it if they've been writing a story about you for long enough. If you can always provide evidence in writing and do so promptly on every attempt at under-bussing that you are aware of (keep that in mind), the narrative that starts to form will be about the weasel and not you.
As a developer at a fairly hectic/chaotic "fast-paced" ad agency, I'm personally very careful to make sure any promises I make are both stated and ideally acknowledged in email in case somebody tries to rewrite the narrative when it turns out something needed to be done much sooner or differently than it was handled which tends to happen a lot, often out of ADD shenanigans more than any real malice.
And learn to keep an eye on anybody who never wants to commit or acknowledge anything in email. They might just hate typing. Or they might be slippery weasels. Sadly you can't ignore politics. I tried for years. If you conduct yourself professionally the best defense is to preemptively strike the crap out of any attempt at sullying your reputation. Using email well provides plenty of ammo for that.