[no spoilers until they're marked]
Since I had to rant about the things that I dislike about FY, I think it's only fair that I comment on what I do like (although I did some of that in the process). So here goes...
Characters and their relationships are the main strengths of the series. The characters managed to draw me in, which is why I stayed. They grow throughout the series, and their interactions and relationships are enjoyable. They are characters who I can care about. The villains aren't intrinsically evil; they have been hardened by the wartorn area that they grew up in, a product of their surroundings.
There are definitely elements of the plot that annoy me, but it's generally intriguing and well-constructed, and the twists are wonderful (episode 27-28, my favorites, come to mind). I don't find the overall plot of the last five manga (second OAV series, I suppose) as good as the rest, although character interactions are fun to see.
I like the humor. That's one of the main things that sets it apart from a soap opera. At the same time, I do enjoy some of the melodrama.
Of vital importance is the fact that Fushigi Yuugi has the ability to make me sad. Any story that has the power to thoroughly depress me is good.
There are tragic but uplifting moments, of course, particularly at the end of the anime series. Miaka sitting on the pavement and proclaiming to Keisuke that the book was the best she'd ever read comes to mind (I like the surprising ending, BTW--the anime ending's much better than the manga ending, IMO). And the slow version of "Itoshii Hito no Tameni" suits the tone of most of the series.
I'll write more/elaborate when more comes to mind and I have more time...