Trying to find what the Whites (Blues, Greens and Red...) racing factions stables might have looked like, I ended up with these results from a Google images search and figured I'd share them to help those of you who also like visuals. (If I was an artist, I'd attempt my own illustrations, but I'm not!)
While not racing stables, I did come across this neat article by English Heritage about cavalry stabling, which may be of use to anyone whose character is or has been associated with cavalry at any point.
I imagine that when complete and in use, they may have looked a little like the Royal Stables of Cordoba in Spain:
Bonus pic: The Circus Maximus as the Romans knew it:
Videos of the Circus Maximus: