Something being a "right" doesn't mean it's a magical force that overrides all others. Rights are always in tension with other rights. A society deciding something is a "right" is a statement about priorities and values.
On the medical care example, most countries have something like the US's EMTALA -- not a Constitutional right but but a de facto "right to medical care" in emergency rooms and some other carve-outs.
But the fundamental right to medical care this codifies doesn't just override all other things. You'll get rightfully and legally-correctly turned away from an ER if you are extremely violent and belligerent, and cannot be reasonably stabilized enough for everyone else in the ER to be safe. If you're in a remote area and need a airlift, but the weather conditions are not reasonably safe for the only helicopter crew available -- as determined by themselves -- then they're not obligated to rescue you, either morally or legally.