There. Go. Apply. Put together a resume. Tell them why you want to work for them. Tell them what you've done. Email them. Hell, point them to this thread, about how they were the one company you could think of that you want to work for.
Let them decide if you are qualified.
The smartest people I know claim to know the least. It's that double-edged sword: the more you know, the more you know you don't know.
If you want, I'll take a look at your resume and cover letter. My email address is in my HN profile.
But seriously, if you want something, go for it.
Don't be the 'no' in your life.