Getting yourself flat on the floor of a free-falling elevator could be interesting, as you will be experiencing zero-G at the time.
You could wait for air resistance on the elevator to take effect as it accelerates, but I imagine that its terminal velocity would be quite high so this might take a while.
Depending on the failure, you could fall up rather than down since the elevator is counterweighted. If the cable breaks, the car will fall down. However, if the drive train breaks in a car that is loaded less than the counterweight (very likely), the car will likely fall up since the counterweight will weight more than the car. My MechEng prof in college was involved as an expert witness in a case of this: the bolts holding the drive pulley sheared, letting the pulley free-run. Bad.
There are almost no cases of an elevator's cable just snapping and going into downwards free fall - with the exception of (possibly) during 9/11 and an incident in 1945 where a bomber pilot rammed a building. The elevator operator dropped 75 floors but miraculously survived:
> One of them fell from the seventy-fifth floor with a woman aboard—an elevator operator. (The operator of the other one had stepped out for a cigarette.) By the time the car crashed into the buffer in the pit (a hydraulic truncheon designed to be a cushion of last resort), a thousand feet of cable had piled up beneath it, serving as a kind of spring. A pillow of air pressure, as the speeding car compressed the air in the shaft, may have helped ease the impact as well. Still, the landing was not soft. The car’s walls buckled, and steel debris tore up through the floor. It was the woman’s good fortune to be cowering in a corner when the car hit. She was severely injured but alive
I would hesitate putting my head against a flat surface, maybe sitting down would be a better alternative assuming that a few broken bones are OK but serious head trauma is a lot worse.
I was thinking the same, but you'd have the issue of spinal compression to deal with. A 10G force could easily fold your spine in half if unsupported. Even if you sit against the wall, you're going to experience a significant amount of spinal compression. Think of someone dropping a 100 lb weight on your head.
I think a good solution would be to cross your arms and place them under your head. I'd gladly suffer two broken arms to cushion the impact on my cranium.
If you look at x-gamers when they fall, they are trained to land on their side - I would think that you would want to do the same here for just that reason. Probably with your arms on either side of your head. You still don't have the spinal compression, but you don't risk your head getting slammed.
You could wait for air resistance on the elevator to take effect as it accelerates, but I imagine that its terminal velocity would be quite high so this might take a while.