A category $\mathcal{C}$ is said to be locally small if for objects $A,B \in \operatorname{ob}(\mathcal{C})$, the morphism class $\hom(A,B)$ is a set; a set being a class that is a member of another class, i.e. $X$ is a set if there is a class $Y$ such that $X \in Y$.
Since $\hom(\mathcal{C})$ is necessarily a class, and $\hom(A,B) \in \hom(\mathcal{C})$, it follows that $\hom(A,B)$ is a set. So how do we structure a non-locally small category?
As pointed out by @Stefan in his comment, your mistake is to consider $$\hom(A,B) \in \hom(\mathcal{C})$$ instead of the correct relation $$\hom(A,B) \subset \hom(\mathcal{C})$$ This condition is not enough to guarantee , in general, that $hom(A,B)$ is a set, that is why we need an extra definition/conditin, if you want to obtain certain results (eg Yoneda lemma).
In general, $hom(A,B)$ is a class.
What is true, though generally not emphasized in textbooks, is that objects and morphisms are sets, since they are members of the respective classes $Ob(C)$ and $Mor(C)$