No. At this time, the only playable races are:

Horde: Orcs, Trolls, Tauren, and Undead

Alliance: Humans, Dwarves, Night Elves, and Gnomes

Can I increase the size of my bank?

Yes, you can purchase additional bag slots for your bank. After purchasing a slot, simply place an empty bag in the new slot and it will act as an extension.