That's what safeties are for! We already have a lot of engineering around making elevators safe. TLDR the elevator will clamp on to the rails to slow the free fall.
Neat info below on snapping elevator cables and some of the things that would help you.
https://science.howstuffworks.com/science-vs-myth/everyday-m...
What rails? They're not going to run more than one of these difficult-to-build structures up for each elevator. There will be a single cable, and the car will climb it. What happens if the cable snaps will depend on how far up the car is. Low enough, and it will descend to the surface on a builtin parachute. High enough, and it will perform some portion of an orbit, before descending on its parachute.
Once you have a space elevator, it’s relatively easy to send up some rockets and demolition explosives that just sit on the cables at appropriate locations and stay there waiting for a disaster; should the cable be broken by malice, accident, or natural causes, the explosives can cut the falling cable into pieces small enough that the rockets can safely guide to the ocean (or orbit, whichever has the lowest delta-v).
The travel pods, likewise, can always have rockets on them whose sole purpose is to get the passengers to safely. Yes, it means you’re taking up more mass than you strictly need to, but the great thing about a space elevator is that the mass is no longer a hyper-critical parameter.
Presumably you could have redundant cables. But two almost adjacent cables would still be exposed to many single points of failure, especially any caused by a deliberate attack on the system.
And just a cable falling could be a significant disaster by itself.
The rails? The car will travel over a stationary structure, not be pulled up by a pulley like a conventional elevator is. So it is failure of that stationary structure that we are talking about.
