Anchor one end of the cable to the side of the moon that faces the earth. Drop the other end of the cable to earth so that it drags along the ground. The earth's gravity and friction contributes a degree of tension to the cable. As the earth does its daily rotation and the moon does its monthly orbit the cable drags around, flicking over mountains and knocking over buildings.
Because the earth's rotation and moon's orbit happen on different planes with different periods, it never goes to the same place twice. Everyone within 30.12 degrees of the equator (the earth's axial tilt plus the moon's orbital inclination) eventually gets access to the space elevator whilst simultaneously having their cities destroyed.
People build carriages to intercept and attach to the rogue elevator cable and ascend into space.