Here is a variation on the same theme as Joel David Hamkins' answer.
Theorem. The following are equivalent over ZF set theory:
- Every set admits a linear order.
- For every set $X$, there is a function choosing for each subset $Y$ of $X$ a total preordering of $Y$ which is nontrivial unless $|Y| \leq 1$.
By preordering, I mean a reflexive transitive relation on a set; a total preordering is one where any two elements are comparable (perhaps both ways). A total preordering is nontrivial if there are two elements that are not comparable.