The Patrician's personal working office to which he periodically invites civic leaders for informal little chats.
In a less prosaic and literal part of the Multiverse, a similar nerve-centre of Government might be called the Oval Office...
The View From The Top:
From evidence in Going Postal (pp 66-67), the Oblong Office is most likely situated on the Turnwise Broadway side of the Palace, as Vetinari is able to see the Tump Tower from the window. We are told the view of the City is one of the very best; this suggests the O.O. is on an upper floor of the Palace, perhaps one of the highest. (Vetinari's bedroom is next door and connects to a dumb-waiter system; from Feet of Clay, we know it to be quite a few floors up as Vimes sends a Dwarf up in the lift to nail shut access doors on lower floors.) In Making Money, Cosmo Lavish has scale models in his study that exactly replicate the view from the Oblong Office window. There is an exchange somewhere between Vetinari and Vimes, where Vimes is invited to look out of the window and comment on what he sees. There is a similar window-related discussion between Vetinari and Mustrum Ridcully.