So I think it's possible just about anywhere in the world, if you're ready to do a lot of things yourself (don't rely on services such as a car/bike mechanic, fix plumbing, house issues yourself, don't pay a lot of money on going out etc).
Anywhere with free healthcare.
And I think that's my only interaction with the medical establishment for the last 6-7 years. The one before that was for some document I needed for my driver's license.
Of course, here in the Netherlands you can't avoid having health insurance, but it's technically free for low income people and not that expensive otherwise. In Romania, and I presume that a lot of other places, you can.
So is it that hard to stay uninsured? I do have some gripes to pick with the medical system, but yeah, I do think taking health into your own hands is a better choice (and a lot more is psychological in nature than the medical system would suggest, which is focused on symptoms and not causes).
I remember in my younger years, I would get a cold probably twice a year on average, and I would always get prescribed antibiotics by an overzealous GP. All of that stopped all of a sudden, and I haven't taken antibiotics in about a decade. I've caught a cold, maybe once every two years, but much lighter and goes away by itself in 2-3 days at most.