I am 30 but also a new fan of TOS. I mean I watched it before but I only became really engrossed with the series recently. Not exactly sure what caught me now that didn't before. I always liked science fiction/fantasy and classic film and television (in fact compared to some of the things I enjoy Star Trek isn't even that old) .
I also never minded dated effects as I think story is more important. I also feel today's effects are kind of boring anyways because it is all done by computers. Of course it is going to look more impressive & real but there is less imagination behind it too.
And yes even some elements of the story might seem dated too only because by now we have seen these same concepts so many times. But taking at the time that Star Trek was made those story elements were fresh and new. Looking at the original series in a historical context it is by far the most interesting to me for this very reason. Not that the original series was perfect and some of it's issues were due to the limitation of 1960's television, but for 1960's television it was a pretty creative and original show. It also had a very hopeful outlook for the future (which is definitely different from a lot of science fiction).
Finally though what probably pulled me in is I really just love the main three characters, how iconic they are, logic vs emotion, and their friendship. I am especially partial to McCoy. As a kid I was familiar with Kirk/Spock but somehow I overlooked Bones. Not sure why because now I absolutely love him.
So bottom line yes not only do I really like TOS, it's my favorite.