The Title of the video is misleading. What Boaz Almog is demonstrating is not at all levitation. The disk is not pushed away from the magnet, and it does not levitate. It is fixed or trapped in the magnetic field, by an effect refered to as "Quantum Locking" or "Flux Pinning". This can be demonstrated by locking the disk in space underneath the magnet, which works just as well. If the magnet was pushing, – and so 'levitating' – the disk, it would be a force acting upon the disk to neutralize the gravitational pull from earth, and it would not be possible to fix the disk under the magnet (it would fall down to earth).
The disc is essentially a very thin superconductor. Superconductors will expel magnetic fields from within, but in this case, where the superconductor is so thin, millions (or billions) of single magnetic strands penetrates the disk, allowing for the disk to be locked in place at various distances and angles from the magnet.
The disk is able to carry the cooling material put on top of it (and can carry 70.000 times it's own weight), but to scale the disk up wouldn't simply be a matter of making it thicker (I believe the magnetic strands would no longer penetrate the superconductor). Maybe this can be achieved by layering many super thin discs?
Anyway, say that the experiment could be scaled up, to carry a car, then no, the person underneath would not be crushed – no more than you're crushed by a bridge, walking under it, as it's fixed in place. And you wouldn't really "lift the car" with the hand - no more than you're lifting a bridge, by placing you hand under it pressing up. You can't lift it, and if the experiment had been scaled up to this level, you would also not be able to displace the locked super conductor (with or without a car on it) with the strength of your arm.
This video visualizes this pretty well:
If the superconducter is not so thin, you get another effect, where the superconducter can be 'trapped' at a fixed distance to the magnet, as seen here: