[Aerkuld]

My Dad is an engineer who used to design elevators. He has shown me safety brake mechanisms as that was part of what he used to test. Not only does the winding gear have a safety brake, but there are safety brakes on the vertical guide rails too that will clamp up on the rail and stop the elevator if the cable breaks. Of course, depending upon where the cable breaks you might not have to worry about the elevator falling to the bottom of the shaft but the end of the cable on its way down to you.
These things are tested and tested and tested to make sure they are super safe. On top of that I believe he said that the average elevator service life is something like 3000 hours, after which I'd think the major systems are replaced. Not much chance of anything going seriously wrong really.