The classical motion of gliding dislocation lines in slip planes of crystalline solid helium leads to plastic deformation even at temperatures far below the Debye temperature and can affect elastic properties. In this work we propose that the gliding of dislocations and plasticity may be the origin of many observed elastic anomalies in solid4He, which have been argued to be connected to supersolidity. We present a dislocation motion model that describes the stress-strain-curves and work-hardening rate d/d of a shear experiment performed at constant strain rate in solid helium. The calculated d/d exhibits strong softening with increasing temperature owing to the motion of dislocations, which mimics anomalous softening of the elastic shear modulus. In the same temperature region the motion of dislocations causes dissipation with a prominent peak.