Imagine how you would feel if a 50-year old guy (yes, an older man; it's the closest we can get to replicating the gender dynamic here) started following you around and offering to take you to the movies. Chances are, it would make you rather uncomfortable, yes?
Even leaving aside the awkwardness of being propositioned by a stranger, you have to understand that gender roles play a very large role in our interactions. As a rule, us guys are bigger and stronger than girls; that immediately means that if anything goes wrong, a girl is at a disadvantage, and is sadly something most girls are forced to be aware of their entire lives. Having a strange guy saying how much he loves you is going to be raising a lot of red flags, and honestly, any sensible girl is going to nope the hell out of there.
So... what do you do?
Listen to Shadowgandor and Vector, they've given you some good advice; meet the girl at a party or outing with mutual friends. Ideally, get the mutual friends to directly introduce you. What you have then is a situation where:
a) You have a reason to approach her; you're no longer some random creep, you're so-and-so's friend from high school. It means there's an icebreaker for the conversation (which makes things easier, and you'll look better for it), and your friendship with the mutual friend is implicit support for your moral character.
b) She's not alone/at a power disadvantage; she's going to be with friends, and she knows she'll be able to say no without repercussions. While that sounds bad, it's not; knowing you have an option to do something is not the same as wanting to do it, and knowing that there is a safe out along the way will actually make her more likely to talk to you. Using an analogy to think about this; you wouldn't walk across a 10cm wide beam strung between the 50th floors of 2 buildings... but you'd happily try and balance on the concrete curb at the edge of the street. They're the same width, but if you fall off the curb, you can safely land on your feet.
c) Parties are social lubricant. People are happy, and more outgoing at a party than they would be in day-to-day life. Unlike talking to her at work (where she may be stressed out, under the hammer, and tired), she will be in the mood to have fun and meet new people. Sometimes, there may be liquid assistance as well... while I'm not advocating getting either you or her drunk, one drink may help you and her relax, and chat more freely.
That said, when you do get the chance to meet her in a comfortable setting... don't be afraid. You do have to be able to go up to a girl and chat. Make her laugh, smile, etc. If you keep sitting in the corner trying to catch her eye, you won't be romantic, you'll be creepy. Don't ask for her number straight away, but if you've been talking for a good half hour or so, asking for her FB details is generally okay. I think it's maybe because it's become a bit of a cliche, but asking for a girl's number has a very clear meaning, and doing that early on may make you come across as a bit of a sleaze.
If you've set on the number though, a good way to ask for it is to find some common interest, and talk about making plans later; I've had good luck with something like this "Ohhh, you like hiking too? A few of my friends go every few weeks... tell you what, give me your number and I'll shoot you a text the next time we are thinking of going."