A car will continue to start and run even with a bad oxygen sensor but the car will run rough or be hard to start or may stall or run rich.
The oxygen sensor is important to a properly running engine and to let the engines computer know what the oxygen levels are in the exhaust to adjust the fuel injection and tell how much fuel should be getting to the engine.
A bad oxygen sensor can cause your vehicle to use more gasoline and cause a loss of engine power as well but it should not cause your vehicle to not start however it may stall especially if you haven't gotten the vehicle running up to temp first.
A bad ignition coil or ignition coils, bad spark plugs, low fuel pressure bad ignition module or the timing being off can cause the vehicle to fail to start.