The flight instructor reported that, during an instructional flight in a twin-engine airplane, which included feathering and unfeathering the left engine in flight, the left engine would not restart. The airplane was unable to maintain altitude and reach the destination airport, so he landed the airplane on a field about 1/2 mile from the airport. After the instructor successfully started the engine on the ground, the student pilot walked back to the airport, while the instructor decided to attempt an off-airport departure from the frozen ground with 4-inch-tall alfalfa stubble. He chose a takeoff path that didn't have any visible obstructions, which he estimated would allow for a 3,000-ft-long ground roll. He then taxied to the edge of the field, held the brakes, applied full power to the engines, and released the brakes, and the airplane began to move. About 700 ft into the ground roll, the airplane impacted a shallow rut from an irrigation pivot, and the nose landing gear collapsed, which resulted in substantial damage to the forward fuselage. The pilot reported that his decision to take off from the field was poor because he made it without first walking the takeoff terrain.