I grew up on a farm in the USA and will try to explain why we use the prepositions at and on with farm, but not in.
On is used for a surface.
At is used for a point.
In is used for an enclosed space.
ON: In your mind, think of a farm. You see the farm out in an open space and it covers / extends over an area of ground / the surface of the earth.
We live on a farm. There are many cows and pigs on our farm. We grow corn and soybeans on our farm. (Our farm = our area that covers / extends over the surface of the earth.)
AT: A point / where.
I will meet you at the farm.
FrauSue�s example: There is really poor phone reception at the farm.
Note: AT and ON can sometimes be used interchageably with subtle differences in meaning:
Days are always busy at/on the farm.
IN: In is not used with farm because the traditional farm is not in an enclosed space.
I can�t think of any instances where in is used with farm. Can anyone else?