I'm at a small undergrad only university. I don't go out with my students to drink / eat / hang out, but the reality is a bit more complex, and has the potential for missteps. Especially because practically all my students (at least, the good ones who I work with on any consistent basis) are women.
Last summer, I ran a field research project where student were enrolled. We did a public day on a weekend, and several students volunteers to help. After it was over, I took them (all of whom were > 21) to lunch and paid for their food / beer. I would never do that in a general situation, but I felt that I owed them for spending their Saturday in the rain working instead of being off doing other things.
Several of my students are older / non-traditional, and so although the basic rules of conduct about professor / student interactions mostly apply, I feel a bit more comfortable conversing with them (in situations like the above) a little more freely than I would with a 20-year old.
Many of my students deal with (or at least are willing to talk about) various issues stemming from mental or physical health, family background, or other things. I probably know more about most of their personal lives than any of my professors knew about mine when I was in school, but I figure if it helps me be effective for them, then it's all right.
Several of the students I work with closely will often come by to chat, sometimes about class stuff, but the conversations have occasionally gone for an hour or more. I have resisted asking them to leave so I can get work done, in part because it usually happens late in the afternoon when my mind is not well-suited for intensive work, and in part because they're nice people and I enjoy talking to them.
In general, the students I work with regularly seem to like me, and it probably helps that I'm fairly young and don't tend to be all that serious. So they tend to be more open with me, and I tend to be more open with them.
I worry that because they know I'm not married, that potentially makes me a possible target for inappropriate crushes, etc., but a crush un-acted on is just a crush. And if they do better in my classes because they want me to think highly of them, I don't see that as a problem.
My university has pretty strict policies on faculty / undergraduate relationships, so the hardest thing (potentially) about keeping things casual and being friendly with students is that among adults who work together regularly, feelings (real or imagined) can sometimes develop. So it has the potential to be a minefield, and although I'm not looking forward to getting older, sometimes I think it might be easier if they saw me more as a father figure than as a male cousin (which seems to be how some of them view me).