Meaning of the Third Amendment
The US Third amendment means that if the US ever wants to quarter soldiers in private houses (or other dwellings), without the consent of the owners, it can only do so in time of war, and only in accordance with some law that Congress passes, authorizing such action and providing procedures for it. To the best of my knowledge, Congress has never passed any such law, and so no such "quartering" would be permissible, even during wartime.
Note that the text of the amendment says "house". This provision might be extended to other dwellings, such as apartment buildings, but probably not to any "privately-owned building" that is not a residence (that is, was not previously a dwelling). Such non-residences are not protected by the Third Amendment , although the takings clause of the Fifth might apply, as might the Due Process clause, depending on the circumstances. Youngstown Sheet & Tube Co. v. Sawyer, 343 U.S. 579 (1952) would be relevant in such a case.
Engblom v. Carey
The amendment was invoked and interpreted in Engblom v. Carey, 677 F.2d 957 (2d Cir. 1982). This is said by Wikipedia to be the first decision at the circuit court level interpreting the amendment or depending on its provisions. It has never been the basis for a US Supreme Court decision
In 1979 there was a state-wide strike of NY corrections officers (prison employees). NY National Guard troops were deployed by the state. Staff had been rented apartments as part of their employment agreement.
Guard troops were first housed in administrative buildings and school buildings at or near the prison. But after five days striking officer were evicted by the state, and Guard troops were housed in dorms so vacated.
Two of the officers sued. The second circuit court held that:
- the National Guard troops are in fact soldiers under the Third Amendment;
- the Third Amendment applies to state as well as federal authorities, i.e., is incorporated against the states; and
- the protection of the Third Amendment extends beyond homeowners, but covers anyone who, within their residence, has a legal expectation of privacy and a legal right to exclude others from entering into the premises. That is, tenants and other lawful residents are protected.
Issues Mentioned in the Question
What law is being referenced?
No such law currently exists. Congress could pass one in the future.
During a war can American soldiers commandeer my house?
Only if Congress passes such a law, and then only if whatever procedures that are "prescribed" by the law Congress passes are followed."