Alright, bro. Here's the deal.
You said you already messed up with the whole eye contact thing. This hurts a lot, not because you messed up, but because this is her first impression of you. If she really does think you're some sort of stalker or something, you're going to have a little more trouble getting out of that hole.
And OH YEAH, I definitely know what you mean about girls that are just completely 'out of my league', as some people put it. It's hard not to be intimidated... but also consider that chances are high that you'll have to initiate everything. Instead of staring at her, try walking up to her and making conversation. A girl's gotta know you at least a little bit before you can ask her out. Don't try too hard... just be yourself and be kick-back. That way, if she starts liking you, good! If she doesn't, well, its better than being liked for something you're not. Just don't be too desperate. I think that girls hate that.
Remember, the first step is always the hardest. Getting something in motion is harder than keeping it in motion. Good luck.
If a dream comes true... then is it still a dream?