Do oxygen sensors eventually all fail?

Sensors tend to last somewhere between 100k and 200k miles depending on operating temperature, fuel contamination, and how sensitive the car is to sensors that are getting old and responding slowly.

OBDII cars (since 1996) have either two or four sensors, from what I have seen.

Generally, the front one(s) fail(s) first, but I replace them all as preventative maintenance when one starts to fail.

As others have noted, a car will run without an oxygen sensor. However, when a sensor gets so bad that the computer ignores it and switches to “limp home” operating mode (check engine light will be on) the car runs rich. That will heat your catalytic converter more than normal and it will devastate your gas mileage. That is important when you are driving across long stretches of desert where there are no gas stations.