Nematic liquid crystal elastomers (LCE) exhibit unique mechanical properties, placing them in a category distinct from other viscoelastic systems. One of their most celebrated properties is the 'soft elasticity', leading to a wide plateau of low, nearly-constant stress upon stretching, a characteristically slow stress relaxation, enhanced surface adhesion, and other remarkable effects. The dynamic soft response of LCE to shear deformations leads to the extremely large loss behaviour with the loss factor tanδ approaching unity over a wide temperature and frequency ranges, with clear implications for damping applications. Here we investigate this effect of anomalous damping, optimising the impact and vibration geometries to reach the greatest benefits in vibration isolation and impact damping by accessing internal shear deformation modes. We compare impact energy dissipation in shaped samples and projectiles, with elastic wave transmission and resonance, finding a good correlation between the results of such diverse tests. By comparing with ordinary elastomers used for industrial damping, we demonstrate that the nematic LCE is an exceptional damping material and propose directions that should be explored for further improvements in practical damping applications.