Just for fun... it might not be impossible:
You can reduce R^2 to {x | 0<=x<=1}[1] and thus, by induction, we can reduce R^n to {x | 0<=x<=1} for any integer n.
If I remember correctly, there are about 10^80 particles. The position of each particle is a point in R^3 so the position of all particles is R^(3*10^80). So, the position of every particle could be stored by the position of a particle on a 1 meter (foot, inch, whatever) long stick.
Of course, you run into problems if space is discrete or, in any event, with the Heisenberg uncertainty principle but you can still store a lot of information with each particle.
Horribly impractical of course, but like I said, just for fun...