Specifically non-alcoholic, but I'll leave the title as is in case someone has the same question for alcoholic root beer.
1Good question. Hard cider has been deemed on-topic for a variety of reasons I wouldn't have anticipated, but in this case perhaps the "overlapping audiences" and other such arguments don't apply as nicely, so I tend to agree with @LessPop_MoreFizz.– Andrew CheongMar 23, 2014 at 21:07
Root Beer doesn't really have anything in common with actual beer, other than the name, and a tiny amount of yeast (and with it, from some processes, a tiny amount of alcohol as a result).
And even the name itself was mostly the result of a marketing gimmick by its inventor.
So, my inclination is to say no to Root Beer and Ginger Ale, given the current scope of the site.