Memorable Kirkjubær is a tiny church dating from 1925 but now in private hands. It has been renovated into a cute one-room guesthouse; the pulpit and altar are still there, and some of the pews are now part of the furniture. There’s a kitchen and bathroom, and the beds (four doubles) are on the upper mezzanine level, with minimal privacy.
The best option would be for a family or small group to hire the whole place out (price negotiable). You can also rent a kayak or rowboat from the owner.