... after a certain point, it may become obvious that some people just want to complain, or to be angry, or to be an asshole, or whatever, and that nothing a reasonable person can do will ever make those people happy or satisfied. So you give them a quarter, metaphorically or otherwise, and tell them to call someone who cares. Because you have other things to do. And then you go on doing those things you need to do. They won’t be happy, but then they were never going to be happy, and it’s not your responsibility to fix their problem — “their problem” not being whatever specific complaint or grievance they might have, but a worldview that requires them to always have a complaint or grievance, and/or to believe that the root of that complaint is somehow about you. That’s something for therapy, perhaps, not for you, or anyone else who isn’t getting paid by the session.
One of the toughest things to swallow about Buddhism, and one of the things I think that turns a lot of people away from it (mostly people who, from what I see, have no idea what they're getting into when they start studying it in earnest), is that one of its implied lessons is that if there's a problem, it's always your problem.
This is tough to get right. It's not meant in the sense that you're always going to be an irreparably terrible person or a miserable sinner — if so, why bother? — but that the root of every issue you're going to have with the universe is always somewhere in your mind.
Sometimes those complaints are legitimate; sometimes they are not. The only person who will ever truly be able to figure out which is which is, well, you.
None of this means anger itself is illegitimate. If people weren't getting good and cheesed off about things right now, I'd wonder what the sand they current had their head buried in tasted like to them. Anger isn't legitimate or illegitimate; it's about what's around it.
What it does mean is that your anger is not you, and that people who spend a lot of energy identifying themselves with their anger and not much else are beyond helping, and generally not worth spending much time around either.
If someone has been personally wounded by something I've done, I'm not above trying to sincerely redress that hurt. (I also know I can't make you believe that just by me saying so, though, but I think most people who know me well would agree.)
What I'm not interested in, and what I won't accept, is if someone's grievance with me is existential — that they're not really mad because I did something in particular, but because the fact of my existence is giving them an excuse to get steamed. Nothing I could ever do would satisfy. Such people are in the habit of identifying with their anger — or at least acting like they do; from the outside there's no practical difference — and so any attempts to satisfy them are automatically bound to fail. They're not interested in a fair exchange of anything.
The worst part of situations like that is when you're also dealing with onlookers who think your refusal to engage with such pedantry is a sign that you're a bad person. That, I find, is more the issue than when it's just one person giving you grief. The less prospective audiences there are for such griefers, the better.