I finally found this answer which tells how to use data validation to make sure all the values in one column are unique.
But it doesn't tell if there is a way to write a single validation rule that can make sure that all the values in multiple columns are unique per column. So far the only way I see is to replicate the validation rule for each (of 50 or so) columns individually.
Is there a way to do that with a single rule? It would hinge on where to put the $, I suppose, and the rules for doing that don't seem to be well documented, especially how they get interpreted inside validation rules.