"they took their station by one of the fires in the octagon room"

The octagon room was one of the assembly (or Upper) rooms for society and entertainment in Regency Bath. 

Octagon Room, Bath
Creative Commons Attribution Share AlikeOctagon Room, Bath - Credit: Stephen McKay

The room from which the other assembly rooms branched out, it was often used for the same purpose -- a concert -- that brought Anne Elliot and her party together in this context.  These rooms are still used for similar purposes today.