Suddenly, Advanced Custom Fields that use Select2 (any custom select dropdown, such as on Post Object fields) were broken. Testing plugins one-by-one narrowed it down to The Events Calendar. Apparently version 4.6.4 is loading Select2 and overriding ACF's own loading of Select2, breaking custom fields that use it.

I found a simple fix in the WordPress Support Forums. Then I encapsulated that fix in a barebones plugin you can download here.