That sounds tough, but I suspect you'll have earned a lot more credit than you realise for owning up to the mistake and accepting responsibility for it. As a manager, I can tell you that it's far more common for people to try to make excuses, pass the blame etc...even when it's patently obvious that they have fucked up. I much prefer those staff who are able to take it on the chin, accept it and learn from it. They are the ones who go far in my experience.
You say that you're clearly no good at your current job, but is that actually true? Anybody can make a mistake...we're all human. What is your work like the rest of the time? You are probably wrong that everyone is whispering behind your back, too - most of them are probably just relieved that it wasn't their fuck-up/hoping they won't be next.
How good is your relationship with your line manager? If they're decent, could you talk to them about how you're feeling? They might be able to reassure you that you are good at your job despite this recent blip.
If not, can you hold out for a while and then move? How long is it since you made the mistake? Some reference forms do ask if an employee has been subject to disciplinary measures, but I've only ever seen them ask about the last 12 months. Do you know how long the warning will stay on your file?
Please be kind to yourself. We all fuck up sometimes, and it probably isn't as bad as you think it is.