Definition: For a set $x$ define its cardinality by $|x|=\min\{\alpha\in On\mid\alpha \approx x\}$.
where $On$ is the calss of all ordinals, $\alpha\approx x$ means there is a bijection $f:\alpha\rightarrow x$.
In ZFC we can say that every set $x$ has a cardinality, because $x$ can be well ordered and thus is bijective to atleast one ordinal. Can we turn this around? That is, in ZF, can we construct (find) a set, that has no cardinality? That means $\{\alpha \in On\mid \alpha \approx x \}$ is empty, quivallently there is no well order on $x$.
It depends what you mean.
First, note that (in ZF) choice holds iff every level of the cumulative hierarchy is well-orderable. So if choice fails there is a canonical witness: namely, the first level of the cumulative hierarchy which is non-well-orderable.
But that probably doesn't seem to satisfying. The problem is that we can't do much better: while it's consistent that very natural sets (like $\mathbb{R}$) are non-well-orderable, it's also consistent that choice holds for a very long time - that is, that the least such $\alpha$ with $V_\alpha$ being non-well-orderable is much bigger than any ordinal we actually care about in general mathematics. So you're not going to be able to do better than the above.