There is an example of a model category structure on a preoder of families of sets, defined as folows: $X\leq Y$ iff every element $x\in X$ is bounded $x\leq y_x$ by some element of $y_x\in Y$; families $X$ and $Y$ are weakly equivalent iff $X\leq Y$ and for every $y\in Y$ there is $x\in X$ such that $y$ and $x$ differ by finitely many elements; cofibrant objects are families of countable sets.

The example is also quite easy, however, it is set theoretic in nature. You may find the definitions in

A homotopy theory for set theory, I  A homotopy approach to set theory

There is an example of a model category structure on a preorder of families of sets, defined as follows: $X\leq Y$ iff every element $x\in X$ is bounded $x\leq y_x$ by some element of $y_x\in Y$; families $X$ and $Y$ are weakly equivalent iff $X\leq Y$ and for every $y\in Y$ there is $x\in X$ such that $y$ and $x$ differ by finitely many elements; cofibrant objects are families of countable sets.

The example is also quite easy, however, it is set theoretic in nature. You may find the definitions in

A homotopy theory for set theory, I 

A homotopy approach to set theory