I have often been told that model categories are nice because they give a framework where the construction of the homotopy category does not incur in set-theoretical problems, but I have never been given an example where one has such problems.
Therefore, I am looking for a reference where I could find an example of a category $\mathsf{C}$ with a "nice" class of weak equivalences $W$ but such that the localization $\mathsf{C}[W^{-1}]$ is not a category anymore. Does anyone know a book or article where such an example is given?