I'm a Julia and have always felt quite "lucky" with my parents choice - imo it's classic, very internationally well used (it's in the top 50 for girls in over a dozen countries), simple to pronounce and spell, yet it's fairly underused here in the UK.
I don't think it's dated - unlike similar Julie, it's never quite made it to the top 50 in the UK historically and I've met a few Julia's of very different ages over the years.
I do like Juliet almost equally though - actually that's the most annoying thing about the name to me, as I'd have liked to been able to use Juliet if I'd had a daughter! 