The pilot reported that he was about three miles south of the non-tower-controlled airport, inbound for landing, when he heard another pilot announce they were on the downwind leg of the traffic pattern for the same runway. He elected to perform a 360° right turn to “allow the other aircraft time to clear the pattern.” He further reported that while in the turn, he “noticed acres and acres of pristine/plush land.” The pilot then decided to perform an off-airport landing in a field. During the landing roll, livestock began to walk in front of the airplane and the pilot took action to avoid a collision. This resulted in a nose over and substantial damage to the vertical stabilizer, rudder, wings, and fuselage. The pilot reported that there were no preaccident mechanical malfunctions or failures with the airplane that would have precluded normal operation.