I suppose it depends on the rules - I would think they could pick one up and swing it around but would not get any proficiency bonuses compared to say a warrior class.

There is another angle to consider while I think about it - axes are not technically sharp weapons (sits somewhere between a blunt and sharp weapon when you think about it) so technically might not be against a healers / priests weapon code of 'no sharp weapons' but also would likely offer no magical benefit compared to a staff of some sort.

So might be technically possible but balanced out by being significantly less efficient and/or having restrictive conditions about when it can be done so (say only as a last survivor situation like in the story) - I'm guessing you meant in a table top RPG environment like dungeons and dragons, right ?