In non-equestrian terms, a soldier's living quarters.
In equestrian terms, the billet or billets are the straps on an English or saddle seat saddle to which the girth is secured. A saddle has three billets on each side, although only two are used (the third one is a spare - because the last thing you need, trust me, is your girth coming off mid ride).
Billets often become worn and routinely have to be replaced. Because of this, it's not uncommon on older saddles for the billets (which are hidden) not to quite match the color of the saddle itself.