They ask because they feel guilty about not doing something, even if there's nothing they can do.

If you weren't okay, you might say something like "No, but I don't want to talk about it now," and they might make a point to find you again later when there's more privacy. I usually find that people ask that when someone's looking uncomfortable/unhappy, and like catamorphism said, if there is something bothering you, a friend might feel obligated to try to help you out.

I do find it irritating when people follow it up with "Are you sure?" though I can understand it because sometimes when I'm not too happy about something, and someone asks me if I'm okay, my knee-jerk reaction is "yeah, i'm fine" mostly because I don't know how to bring up whatever topic it is, and they say "are you sure" and I say "Well, actually..."

I think that people are trying to indicate that they care about you and are thinking of you. It's meant as a kindness. I think that in the back of their mind, the idea is that you will feel better merely for the question having been asked.

As for an answer...that's trickier. Maybe asking "Are you okay?" in a mock serious tone? Is there a silly thing you could say in response that'd be natural for you? Other people might be able to get away with "How can I be okay when Bush is in the White House??" but I don't know if that would work for you. (The point being that you want to make the person realize they asked a silly question you can't possibly answer but you want to do so in a way that doesn't dump all over their good intentions.)

I don't think so. That sounds like an effective way to encourage people to stop caring about you.

Reaching out to someone you care about is not silly.

No, no it's not. I think you may have misunderstood me. You don't want to make people feel bad for caring. But you want to make people understand in as nice a way as possible, that while you appreciate their intentions, what they are doing isn't helping. (Or maybe that now isn't the time to discuss it.)

Really--there is no good reply to "Are you okay?" when asked in the middle of a large group of people. I certainly wouldn't admit to not being okay unless the circumstances were pretty bad indeed. This means the only reply possible is "Yes", which make it a silly question, however well-intentioned.

Thinking about it, the replies I suggested for Becca probably wouldn't come off naturally at all for her, and she'd sound more sarcastic than playful, which would have entirely the wrong effect. zdamiana is smarter than me.

Does she think I'm lying?

Probably; Otherwise she wouldn't have said, "I don't believe you." Though perhaps lying is too strong a word in this situation, since, even if she does think you may not be telling the truth about how you feel (and technically, not telling the truth = lying), that is about as benign an un-truth as you can possibly tell. Your not admitting your feelings to someone who is concerned about you does nothing to harm that person. So although a reasonable person in this situation may believe you to be lying, she oughtn't be offended by such a lie. She might legitimately feel regret, for your sake, that you don't apparently feel comfortable sharing your feelings with her, but she shouldn't take offense.

Do I look like something is wrong?

Apparently. People don't generally walk up to other people and ask them, "Are you ok?" unless it looks to them like something is probably wrong, and like the person in question is really not ok at all.

Am I making people uncomfortable?

Doubtful. It seems more likely that you are (unintentionally) making people feel concerned for you.

Maybe I should just leave. Usually I do.

Now that's what makes them feel uncomfortable. Your leaving makes people feel that they must have offended you, or crossed a personal boundary of yours that they had not intended to cross. It makes them question their own actions, and wonder if they have done anything (unintentionally) offensive, when they had just been trying to reach out to a friend who seemed to be unhappy/uncomfortable/in need of support. It makes them feel that they have, in fact, made a bad situation worse for you, since it seemed clear to them that you were not ok when the interaction started, and, since you left, you were clearly even more not-ok at the end of the interaction. This result can be very upsetting for the sorts of empathetic folks who are likely to notice you behaving differently than usual and approach you and ask you these sorts of questions.

Perhaps a response to that question that would make everyone feel better would be something along the lines of:

Her: "Are you ok?"
You: "Yeah, but I'm awfully tired, and feeling overwhelmed by this crowd. I think I just need some quiet time."
Her: "That's too bad. I hope you get some rest."
You: "Thanks"
She leaves you alone.

That way she would walk away feeling that 1) she had been correct in perceiving that you were not entirely ok, 2) although you aren't entirely ok, at least it is a relatively minor not-ok state, and she doesn't have to worry about you further, 3) you apparently aren't in the mood to talk, 4) if you leave shortly thereafter, she won't become all the more concerned about you. Meanwhile, you wouldn't be accused of lying, and the interaction might not get to the point where you do feel compelled to leave.

You don't have to use exactly those words. I picked that response because, knowing you, I thought it might be the most likely reason why you might come across to others as not feeling ok in that sort of situation.

It would probably be best for you to try to figure out what it is about your demeanor at the times when this happens that makes people suspect that there might be something wrong. Then think about what is causing you to have that demeanor. So, once you've given it some thought, a more truthful response might actually be, "I'm fine, I just sometimes have trouble warming up to social situations like this one," or "I'm ok, but I'm getting a little annoyed by some of the conversations I've overheard," or, "I'm fine. This is just one of those social situations that makes me feel a little awkward a first. I'm sure I'll warm up to it, though."

Or, if you have trouble figuring out what it is about your demeanor that is different in those situations, you could always ask:

Her: "Are you ok?"
You: "I'm fine, but you know, it's weird... people seem to come up to me in situations like this a lot and ask me if I'm ok, when, in fact, I really am perfectly fine. It makes me think I must be doing something that makes them think I'm not ok. The frustrating thing is that I just can't figure out what I'm doing to make people think I'm not alright. Can you help me figure it out, and tell me what made you ask me that question?"

You wouldn't lie that way, and the questioner could feel helpful - like she was, in fact, helping you to solve a problem that's been troubling you (which is actually exactly what she set out to do). You wouldn't get accused of lying, and you might gain some helpful insight to help prevent that sort of situation from arising in the future.

zdamiana hit it on the head.

To avoid having the question asked at all, the more difficult thing to do is to pretend everything is okay. The shoulders back, smile forward thing. Seems a little duplicitous but no one asks confident people if they are okay (unless they know them really really well). In new social situations, I used to withdraw, but that made me look lost and then people would approach me and asked if I needed help. I was trying to avoid drawing attention so that wasn't working. If it's a mixer sort of setting, sometimes it helps to go get a drink just so that there's a purpose and then there's a prop to hold. There's nothing wrong with telling the truth too - "I'm feeling slightly awkward, but it's okay."

I've gotten "Are you okay?" a LOT since Evan died. It's the same thing, albeit in a different way. Obviously I'm not okay, and I'm not going to be okay for a while, so I'm not sure what the right answer is.