At the conclusion of an uneventful on-demand, instrument flight rules flight at night, the flight crew of the business jet conducted an instrument approach to a runway that was about 7,300 feet long. The crew noted that the weather conditions during the approach included fog and mist, and after touching down about 1,500 feet from the runway threshold, they applied the brakes and thrust reversers gradually due to the "slippery" condition of the runway. As the airplane approached the end of the runway, the crew applied full braking, but the airplane departed the end of the runway and came to rest on the side of an embankment. During the runway overrun the fuselage was substantially damaged. The flight crew reported that there were no mechanical malfunctions or failures of the airplane that would have precluded normal operation.