Language that is not well-orderable Godel's completeness theorem can be stated as "Every consistent first-order theory with a well-orderable language has a model." (from Wikipedia)
What would be an example of a language that is not well-orderable?

If you assume choice, then every language can be well-ordered by the well-ordering theorem.
Even without choice, any finitary language (i.e. a set of finite strings) can be well-ordered, provided we can well-order its alphabet. 
If we have an arbitrary well-ordered alphabet $\Sigma$, then we can well-order $\Sigma^*$ (the set of all finite strings over $\Sigma$) as follows: First order all strings in the language by increasing length; then order strings of the same length (reverse) lexicographically, by the ordering on $\Sigma$. Since all finitary languages over $\Sigma$ are subsets of $\Sigma^*$, those too can be well-ordered in the same fashion.
Certainly any countable language can be well-ordered, since by definition we can put such a language in bijection with the natural numbers, which are well-ordered.
So assuming the failure of choice, a finitary non-well-orderable language would need to have an uncountable, non-well-orderable alphabet.
