When the word bureau was borrowed in the late seventeenth century from French, it meant "writing desk". Because these types of desks were very common in offices, the word also got metonymically applied to them, and that's how the definition "government office" and the word bureaucracy came about. Those desks also had drawers, which is why North Americans additionally use bureau to mean "chest of drawers". Going backwards in time, it gets even more interesting: bureau comes from the Old French word burel, which meant "woolen cloth" because they were frequently draped on top of writing desks as a cover. That's a diminutive of bure, meaning "dark brown cloth", and ultimately derives from either Latin burrus ("red") or burra, which referred to a type of wool garment.
Adam Aleksic, a rising sophomore studying linguistics and government at Harvard University, has been described as the internet's sixth most famous etymologist. He also has disturbing interests in politics, vexillology, geography, board games, conlanging, and law.