I know the general rule of thumb about not shitting where you eat, and I'm completely okay with the risk.

We work in a hospital together. Not in the same department/floor, so our paths sometimes cross, but we don't really interact too much. It's been a couple weeks since I've seen her last, but we are friends on Facebook. She's an aide/nursing student, so I'm not even too sure when I'll see her again.

We've never really held a super long conversation, but we've talked a few times here and there over the past year.

I want to ask her out, but not sure how I should go about doing it. When she's at work, she's almost always busy... and if she's not, she's always around nurses/doctors/co-workers/patients and the such. So it's hard to pull her aside and talk to her personally.

Should I consider talking to her over Facebook? Or another option I should try?

Well, kinda in the same place as you. I have added and contacted her on FB, today! (such coincidence). Just not giving a fuck about the work stuff really. We are adults and I assume we should handle this situation. At least I can!

To the OP: Avoid contacting her over FB if you can. You said she's "almost always busy" at work; yes she is doing important work but I'm sure she can spare a couple or a few minutes for you to have a chat at some point. If you don't have verbal game chops from routinely opening women it may be an uphill climb to quickly get some sparks going. But if the chat is going well, just tell her you're busy but you'd love to continue this out of the office sometime, see what she says to that.