All questions are related.

How does one go about asking someone to hang out? Everyone likes me well enough at work. But nobody ever says; hey, lets do something after work! How do I initiate?

How do I approach a woman to ask her out? What if we work together. I really don't want to say, "lets go on a date. I like you." I don't want to be rejected again. But I really like her. I've heard the best relationships begin as friendships. I want to make her a friend, not just a co-worker, but how?

She does live far, too. Complications... I hate complications...

How do I hint to her that I am interested?

How do I expose myself like that again?

Starting in the new year, I probably won't be working there anymore. How do we keep in touch? I'm so bad at keeping in touch, and she's a busy person.

She is so nice.

Ok. Sorry. I just had to get all that out. Can you tell I'm not so great at the relationship thing? Yup. Never flirted in my life... as far as I know... :confused:

Hang it all.... Merry Christmas!

You probably don't want to scare her off and you don't want to get hurt again.
It depends where you work; some fields are more difficult to get people to hang out than others. Sometimes everyone is so tired that all they want to do is go home and straight to bed.

Do you know where she stands in terms of liking you the same way you like her?
You don't want to come forward and open up to her when there's nothing there from her side. :( Casual talk is a good way to start and feel her out; no expectations; your instincts should start 'waking up' as of when to approach her or not.
I agree that the best relationships start with great friendships; that is how mine started.
Do you know her outside of work? She could seem nice but only because it is work;outside she could have a different personality.

Long distance relationships can be hard; it takes a lot of patience from both sides to get it going and some people just give up on them. They can be real lessons.
You know she is busy and keeping in touch will be tough; every now and then show up at work with the 'I came to see you' line and see if that works. It did for me.