It isn’t possible for the man to hang himself without having some other means of raising himself off the ground, at least initially. There is nothing else in the barn, and we are told that it is impossible to climb the walls or along the rafters. So there must be something which has disappeared. There are a number of solutions, but the favoured is that the man formed a large block of ice on which to stand, which slowly melted away after his death. The water would presumably have soaked away or evaporated, leaving no trace.