The monastic community of Mount Athos, Greece. It's free in the sense you don't pay anything, but you have to become an Orthodox monk, with whatever that entails
OTOH, any "you can stay indefinitely" offers would be given on an individual basis, contingent on both your behavior and their own situation (including the financial viability of their organization), and rather unlikely to include substantial medical care as you got older.
Anyway I know one person who became a monk at one and another who was rejected for not meeting the requirements. Exceptions have to be granted by the regional bishop, and they need a genuinely compelling reason. "There aren't enough monks" either isn't one, or they don't have that problem right now.