The question is in subject. 


**Update:** See Andreas Thom's answer.



Do separable $C^*$-algebras form a set?
It is not so clear what you mean.


However, every separable $C^\ast$-algebra embeds in $B(\ell^2 \mathbb N)$. Hence, the isomorphism classes of separable $C^\ast$-algebras form a set.


