A billet is a term for living quarters to which a person, generally a soldier, is assigned to sleep. Historically, it referred to a private dwelling that was required to accept the soldier. The abuse of the system under Charles I led to its specific condemnation in the Petition of Rights (1628). It was again declared illegal in 1679 but was practiced under both Charles II and James II. Condemned again in the Bill of Rights (1689), billeting on innkeepers was subsequently allowed by successive Mutiny Acts.
Soldiers are generally billeted in barracks or garrisons when not on combat duty, although in some armies soldiers with families are permitted to maintain a home off-post. Used for a building, the term is more commonly used in British English; United States standard terms are quarters, barracks, "Single (Soldier) Housing" or "Family Housing".
Sources
The Wordsworth Dictionary of British History (J P Kenyon, Wordsworth Editions, 1981)
