"The fathers sit in the office."
Translation:Patres in tablino sedent.
Isn't the office the object?
Not the object of the verb, no. "Sit" is intransitive, meaning it does not take a direct object (you don't "sit something"—well, not in this sense, at least). Instead, what we have is a prepositional phrase in tablīnō describing where the sitting is taking place: "in the office." So the case of tablīnō is dictated by the preposition in which can be found with either the accusative case (tablīnum) or the ablative case (tablīnō). With the ablative case, in means "in" or "on", but with the accusative case means "into."
How do you know that it's tablino, not tablinum in this case?
Because we want to say that they "sit in" rather than "sit into."
"In tablino" is a place, it's a static place. They are already sitting. They don't make the move of sitting, or going in the office, they are already there. "They are in the office" (but on chairs). It's static.
If there was a move, it would be "in tablinum" (to go in the office), or "in sellam (sedere)": the act of sitting.
It's a subtle distinction, but all is about the move here, and the fact to make the move, or to be static in a place.
Just to sum up:
In + ablative = it's the static ablative, indicating something is already there (usually).
In + accusative = shows a move. For instance the fact to move our bottom on a chair, or to go in the office.