Getting yourself flat on the floor of a free-falling elevator could be interesting, as you will be experiencing zero-G at the time.<p>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 <i>up</i> 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 <i>up</i> 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.<p>Interestingly, Otis is famous for his invention of the elevator <i>safety brake</i>. <a href="http://www.ideafinder.com/history/inventions/elevator.htm" rel="nofollow">http://www.ideafinder.com/history/inventions/elevator.htm</a>
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.