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.