The pilot reported that during the takeoff roll the airplane visually appeared and felt to be accelerating normally, but the airspeed indicator never increased above 40 knots. The pilot aborted the takeoff when the airplane was halfway down the runway. The pilot said he used brakes to decelerate but the airplane was skidding on the wet runway. The pilot was unable to stop and went off the runway and collided with a wood fence, which resulted in substantial damage to the right elevator. Examination of the airplane found the pitot tube was filled with debris. According to the operator, the airplane was not equipped with a pitot tube cover.