For what it's worth, from a guy twice your age, the greatest regret I have in life, and I don't have many, is that I didn't take more risks, especially at a younger age. I'm back in the dating scene for the second time, being recently divorced after a long-term marriage. I totally get how you feel, I felt the same at your age, but right now, I'm having the absolute time of my life, mainly because I couldn't care less about rejection anymore. I love to flirt. I do it every chance I get. It leaves a lot of my friends slack-jawed, as though I'm one of those guys that has a special way with women. Nothing could be further from the truth; I'm just not afraid to take the chance.
Learning to cope with rejection and other negative situations is a very valuable life skill - it will help you immeasurably in your career and personal life. One of the things I've found as I've grown older is that there aren't a lot of people out there who really have any balls, and it's amazing sometimes what you can get, just because you're the one guy who had the courage to ask. As Nike says, "Just Do It." I mean, what's the worst that can happen if a girl says "no". You don't self-immolate; you both move on with your lives and go on to the next opportunity. No harm, no foul. No reason to even feel bad. It doesn't mean she doesn't like you; it just means she's not into you enough to date you. Or she's got a boyfriend. Or is recovering from one. Or is just too busy right now. Whatever. I'm sure there are girls that you'd be less than thrilled to go on a date with if they asked, or times when, for whatever reason, you don't feel like going on a date either.
And don't put so much pressure on yourself, or her. "You know, I really enjoy your company; do you want to grab a drink or a cup of coffee sometime?"
If she says "no, thanks", then you smile and say, "It's cool; just thought I'd ask." And the conversation goes on. A little bump in the road. No need to even feel awkward. Just a little, normal human interaction. Even the ones who turn you down often find you endearing for trying, if you have a little class about how you act when you ask, and especially if they say "no" and you're cool about it, and you never know what might happen in the future. You go on with the conversation, each the same people you were before.
Good luck and have fun!