You are looking for a definable class $C$ such that (ZFC-provably) neither $C$ nor its complement can be a set. There are lots of those. Note that a class is not a set iff it contains elements of arbitrarily high rank. If you define a class at random, I would think the chances are at least 99% that it has this property.
Some examples.
- The class of all sets containing your favorite set $s_0$. (As an element. Or, as a subset - unless $s_0=\emptyset$.)
- The class of all groups. (Or, your favorite class of structures, unless it happens to be the class of all sets.)
- The class of all finite sets. The class of all sets of size $\kappa$.
- The class of all sets of even rank.
No, this is too easy, I must have misunderstood the question...