In a previous question Joel Hamkins linked to a model obtained by adding Cohen subsets to every regular cardinal. This model had the property that the universe could not be linearly ordered.
In particular, there was a class of pairs that witnessed that. Namely there was a class of pairs without a choice function. Consider now the class of partial choice functions. Of course there is no injection from the ordinals to that class, since from such injection we could have engineered a choice function from the class of pairs.
But on the other hand, of course there is no injection this class into the ordinals, since in that case we could have easily defined a choice function using the well-ordering obtained from such injection.