I'd say that some of it might be theoretically possible, but it's an awful stretch.
Survive a full dose of nuclear radiation
iandotkelly ♦ has answered that in the comments. However, I do have a problem imagining a "cure" that can keep such a person working for the next few days and then he's supposed to just drop dead. Whole of the body decays, and while some zombificating technology might keep the body moving, the brain decays as well. If the cure cas put a stop to that, it should be able to do it permanently. If not, person would soon become delusional, or faint, or something... but would not be fit to work with the heavy machinery.
Store data in brains
Since we use whole of our brains, I'd say that such meddling with it would necessarily affect some (or even most) of its usual functionality. I'd allow it as a distant possibility as I don't know much about brain, but I really see no way to avoid damaging the brain (or, at least, significantly changing the person's personality).
Maybe the trick is that John Carlyle is a corrupt CEO, so he doesn't use most of his brain? ;-)
But why would simple reading of the data destroy the brain, when writing to it didn't do anything? That's just pure nonsense, IMO.
Shoot a missile from Earth out of a moderately-sized rocket launcher and hit a ship flying in space
You'd have to calculate how much energy would such a rocket need. It is certainly not nearly possible with today's fuels. Might it be theoretically possible to create such fuel, I'm not sure.
A machine that is able to fully heal any diseases, even terminal ones, in a matter of seconds
Well, in theory, why not? You'd need to replace bad cells, maybe edit genes,... avoid altering personality,... Time is not important, IMO. If it is possible to do it, then it is probably possible to do it quickly. In theory, at least.
However, notice that we still use stitches to close wounds, i.e., we rely on body healing itself, while we just help. So, as far as I know, there is no technology capable of regrowing anything (although, we do know how to grow some organs in labs).
Reconstruct someone's face perfectly using said machine, including facial hair
If you can do what is mentioned in the previous point, and you know how to read the DNA and decypher its tiniest details, you should be able to reconstruct the face. As for facial hair (or, better, haircut), you'd need a "barber" function in that machine, along with some info on what kind of haircut the person had before.
But, the guy surviving with the hole where his face and a good part of the brain were... that has nothing to do with the technology, and is absurd no matter how technically advanced the society becomes.
I wouldn't suggest watching Elysium a bit more seriously than a popcorn action flick.