Yes, it is possible to rule out certain compactification spaces.
First, it's not "our space" or "our universe" that's a Calabi-Yau manifold. The way we extract four-dimensional physics from the ten-dimensional string theories is by supposing that the ten-dimensional "spacetime" is a product $M\times C$ of our spacetime $M$ (usually either supposed to be Minkowski space or deSitter space) and a six-dimensional compact manifold $C$. Formally, one attempts dimensional reduction from the ten-dimensional to the four-dimensional theory, as a higher-dimensional analog of what is done in Kaluza-Klein theory.
In order for the four-dimensional theory to retain some supersymmetry (which we usually desire), $C$ must be a Calabi-Yau manifold. Note that it is perfectly valid to compactify with $C$ not Calabi-Yau, but generally, the resulting four-dimensional theory won't have supersymmetry, which is not strictly a problem since we haven't observed supersymmetry so far. So, purely phenomenologically, it is not even absolutely certain we should be compactifying on Calabi-Yau manifolds, but since this is a type of compactification we understand much better than the generic case, we study it nevertheless.
"Realistic" model building is a significant part of on-going string theory research. This means that we are still examining various types of Calabi-Yau compactifications for the phenomenology they induce in four dimensions. In particular, the particle content of the four-dimensional theory is usually comparatively "easy" to determine from the geometrical properties of the Calabi-Yau manifold, although this is complicated by the possibility of branes and questions about the regimes in which the supergravity approximation to the string theory and therefore the validity of Kaluza-Klein reduction is vaid. There are many examples which do not yield a phenomenology close to the Standard Model and hence should be considered "ruled out" by now. On the other hand, some models do yield Standard Model-type phenomenologies.