Why model space with real numbers? Why we use $\mathbb{R}^{3}$as a model for space? More specifically why we don't use other number systems such as extensions of the real line (hyperreals, surreals, etc.)?
 A: Superspace in supersymmetry has Grassmann numbers as coordinates. BPS compactifications in string theory have complex numbers as Calabi-Yau n-fold coordinates. Noncommutative geometry has noncommutative algebras as coordinates.
A: The two examples you gave, hyperreals and surreals, are exactly the same as the real numbers as far as physics is concerned. They are logically different from the real numbers, in that they both contain special infinitesimal elements $a>0$ which have the infinite list of properties $a<{1\over n}$ for each natural number n, while the real numbers do not. And this does make them different as a logical model of the axioms of the real number line, but this does not mean that they give a different as a model of space, as we see around us in nature.
A model of nature is always expressed by matching computations that you can perform to experiments which you can set up. The computations are done on a standard digital computer. In principle, you don't even need real numbers, because a computer runs on integers.
The real number models of space are always explicitly defined as a small $\epsilon$ limit of some discrete structure. The limit has to make sense, so that you can compute the small $\epsilon$ limit. In certain cases, the limit has been worked out already, and the theory is defined on the continuum from the start. In other cases, like in quantum fields, the limit still has to be taken by hand on a case-by-case basis.
The reason for using continuum to model space is ultimately because we are big. If you make a lattice model, the description at large distances is often by a continuum limit. Any model purporting to describe physics must use the continuum in such a way that if you take the spacing small, you recover the continuum answer. This is explicitly true of quantum fields, for example.
To show that this is not a completely empty statement: Two particles move through space, and when their distance is exactly rational $p/q$ in lowest terms, then they feel a mutual impulse in the direction of separation proportional to ${1\over q^3}$. This model is absurd as physics for obvious reasons--- it treats space-time points as resolved to arbitrary accuracy.
A: Both the examples you gave are real number systems, they're just a different way to slice up and interpret values. As for why we don't use it, the better question is why would we.
