Single datum: I barely got the words "I quit my job the day before yesterday and am hence..." out before the friendly clerks at city hall put an application for (Japanese) national insurance in my hand. Had a card 20 minutes later.

The 3-month-tourist-visa case is pretty tricky. You can buy travel health insurance pretty reasonably, but most policies limit the length of a covered trip to 1 month at a time, and specify that it cannot be for work or study. If on the other hand you have official residence in another country, you can either buy local insurance there, or be covered under their national system (if they have one). But if you're somewhere for 3 months as a tourist (or "tourist") and have no local ID/registration, getting insurance that will cover that scenario is not easy. It's possible some travel insurance doesn't have the 1-month-per-trip limit, but all the ones I've seen do.

Its mainly a "expect to pay a few grand" kind of deal.

I had 2 spine surgeries and LASIK done this year. Maybe 300k rupees or about $4000. I could have gone cheaper, but at that price I just figured "why risk it"?