The Beta badge is achieved by "Actively participating in the private beta." Does that mean the private beta for Stack Overflow (which I believe never happened) or just any site on the Stack Exchange network?
There was a private beta for Stack Overflow - it was in July 2008!
The badge is site-specific, so you get it for a site when you participate in the private beta for that site. It's way too late to get it for Stack Overflow, but if you commit to a site on Area 51 and ask and answer questions in its private beta, you'll get the badge for that site.