building, mansion, large building, palace
館 centers on a large, important building: a mansion, palace, or any grand structure. It can also refer to a building that houses a specific function, like a hall or inn.
館 combines 飠 (food) and 官 (official), suggesting a building where officials are fed or lodged. This points to a large, official building or mansion.
The left side 飠 is food, and the right side 官 is an official. Picture a grand building where officials gather to eat: that's a mansion or palace.
For カン, imagine a can of food being served in a grand mansion: can -> カン, and the building is where the feast happens.
library
museum
art museum; art gallery
embassy
ryokan; traditional Japanese inn
meeting hall; assembly hall
gymnasium
aquarium (building)
inside a library, museum, cinema, etc.; on the premises
public hall; community center; community centre
main building (as opposed to an annex, outbuilding, etc.)
opening (for that day's business; of a library, museum, cinema, etc.)
superintendent; director; curator; chief librarian
consulate
closure (of a library, museum, etc.)
annex (to a building); annexe; extension; supplementary building
reception hall (esp. for visiting state dignitaries); guest house
official residence
admission fee (for a museum, etc.)
national museum
diplomatic mission; overseas diplomatic establishment
movie theatre; movie theater; cinema