This one doesn't seem like it'll be a problem; talked to her at lunch yesterday and she seemed completely normal. I think if I hadn't actually witnessed the semi-breakdown I would never have known.
It's likely that if you make yourself available to her as someone who will listen to and support her, you'll see more of things that she's not displaying to everyone she meets, which
may probably will include more stuff that requires more listening and support. Not that that's bad, just something to be aware of before you get into it.
You don't try to take advantage of her emotional problems to get with her
This is the difficult one. I mean, I don't want to sound like a rapist. Certainly my goal here is to improve the lives of OTHERS, not me. I'm not going to sit down and think "Now how can make this situation end with me having a chick."
That said, what if I give the wrong impression? What if she gets the idea? What do I say then? "Sorry, but I can't date you on account of you're crazy?"
The first step, I think, is to decide if you want to date her, potential crazy and all. I think that's a choice you need to make, not one for The Forums to make, since you're the one who knows her. That said, dating her just because she's a cute girl probably isn't a good idea, since it sounds like deciding to date her will mean committing to a non-trivial amount of emotional support. You likely won't be able to just avoid the subject, as cowofdoom suggests.
If you decide that you're willing to take that on, there isn't any particular reason not to go for it. Talk to her more, show interest in her (in a friendly way, of course), support her, and she'll likely open up to you slowly until you feel ready to more formally "ask her out". Be aware that pushing her to open up more will just slow things down, and making sure that you always make it clear that you accept her for who she is will speed things up. Doing that (even if you do go your separate ways in a few years) will not get you into the special hell[1], so don't worry about that, and assuming you aren't a total jerk you'll likely keep her as a close friend.
If you decide that you aren't, make sure you set firm boundaries. It sounds like you'd like to help her regardless of the dating question, so the first bit of the preceding paragraph is still relevant. The best way to support her is to help her see that not everyone is like her family, and the best way to (initially) do that is to be friendly and supportive to her. Since on this path you aren't aiming for dating, it's likely enough to just talk to her and not demean her. Longer-term, you'll probably want to try to introduce her to other people you think she'll get along with, and then that group can provide support and stuff for her without you shouldering a lot of the load.
Likely, in the second situation, dating will never come up, but if it does a reasonable response is something along the lines of "I see you as a [close] friend, and enjoy spending time with you (which you seem to) and care about you (which you also seem to), but I'm not interested in you that way.". It helps if you can give some other reason (most commonly, "I'm already in a relationship" or "I don't feel like I'm ready for a relationship right now", since those both shift the reason onto you rather than being a rejection of her. In the second case, don't go start a relationship with someone else shortly afterward), but your basic goal is to reinforce the idea that you are not rejecting her as a person, you just don't have dating-oriented feelings for her.
Either way, I hope it goes well.
[1] You'd only really end up in the special hell in my book if you used her bad situation to pressure her into doing things that she didn't want to do but felt like she had to in order to feel accepted/supported/cared for. That doesn't seem like the route you're going for here at all.