I've been reading articles about how to get over a mistake. All of them mention making up for it somehow/learning some lesson. But what if it's a mistake that literally can't be made up for, or there's no lesson beyond "don't be an idiot"? Not a single one of them talks about those. So...am I just stuck?
I unfortunately don't think any of us can necessarily offer advice without knowing what exactly you messed up with. I understand if you want to keep things private, but again, no one can really say anything about your unspoken situation.
Most things can be helped in some way, even if you just learn a lesson from it, assuming you didn't commit some kind of heinous criminal act...
It does depend a lot on the specific mistake. Like, if it was something that resulted in the loss of something precious/irreplaceable, it can be very difficult to make that up. And if it resulted in the loss of a life (such as a pet or even a human being), that can be even more difficult--if not impossible. Sometimes relationships can be destroyed by a mistake as well, and so many factors can go into that, that there is no straightforward answer.
The reality is that people will make mistakes--large and small, inconsequential and life-changing. I've made mistakes that I felt horrible about, and they can still feel a little uncomfortable years later. It does get less difficult to cope with as you grow and recognize that your mistakes don't totally define you. Like when I internally cringe over something I did when I was 15, I remember to myself that I was a literal teenager and didn't have a lot of knowledge or support back then. It's been more than 10 years, and I'm a better person now.
It's important to reflect on what you did learn, though--I don't think there are so many mistakes that boil down to "don't be an idiot." They're usually more complicated than that, and it might take some deeper reflection to see what the true lesson is. Of course, nothing's truly universal, either!
Yeah, I really don't want to say much about it because of how utterly embarrassing it still is for me. As detail-less as possible, it's a should-have-known-better-how-and-why-the-hell-did-I-let-this-happen? school thing, which is why I mentioned "don't be an idiot". I really can't think of any other lesson to be learned from it.
Look, we all make mistakes, we sometimes make bigger mistakes and other times smaller ones. And some of them are more embarrassing than others. It's through mistakes that we learn what's the right thing to do in certain situations.
'don;t be an idiot' is not really a lesson ... what you can learn is to recognize some of the signs/situations that led to this mistake and act in a way that is not the same.
I really have to update my blog ObscureJourney and my review site BeingObscure. French speakers can read my reviews here.
People make dumb mistakes all the time, that's why nobody talks about it; we're used to it by now! and you'll get to that point in your life too. Don't lose your sleep because of those small issues, screwups are a part of the human being and I assure you that you are the only one thinking about this over and over, everybody else already moved on (the reason? they are all focused on thinking and re-thinking their own screwups!) (;

Some things will just always feel embarrassing as all heck, but you know now that you'll make sure it won't happen again, and the more time passes, it'll come up less and less in your mind. Occupy yourself with other things, to put your mind in a different space. It won't go away completely, but it'll get easier to deal with.