They live ____ a farm.
It's definitely on.
A "farm" is an area. You live "on" a farm, but "in" a farmhouse. If you want to live "in" a farm where would that be - on top or under a haystack?
Imagine a couch and a rocking chair with armrests. We sit ON a couch, but IN a rocking chair (imagine an enclosure around us). A farm is a vast area, just like a couch, long. We don't feel 'enclosed' living ON a farm.