How come the Anime didn't cover all 23 volumes? Or is there one that covers all 23 volumes..like an OVA or something?

It costs to make an anime, and at the time they didn't have the funds. There is no scheduled second season or any ova's sorry.

To be quite honest, there are very few animes that follow the whole story of the manga. Unfortunately, that's just the way it is. And no, I don't believe there's a Fruits Basket OVA.