One of the most crucial functionalities of load-bearing biological materials such as shell and bone is to protect their interior organs from damage and fracture arising from external dynamic impacts. However, how this class of materials effectively damp stress waves traveling through their structure is still largely unknown. With a self-similar hierarchical model, a theoretical approach was established to investigate the damping properties of load-bearing biological materials in relation to the biopolymer viscous characteristics, the loading frequency, the geometrical parameters of reinforcements, as well as the hierarchy number. It was found that the damping behavior originates from the viscous characteristics of the organic (biopolymer) constituents and is greatly tuned and enhanced by the staggered and hierarchical organization of the organic and inorganic constituents. For verification purpose, numerical experiments via finite-element method (FEM) have also been conducted and shown results consistent with the theoretical predictions. Furthermore, the results suggest that for the self-similar hierarchical design, there is an optimal aspect ratio of reinforcements for a specific loading frequency and a peak loading frequency for a specific aspect ratio of reinforcements, at which the damping capacity of the composite is maximized. Our findings not only add valuable insights into the stress wave damping mechanisms of load-bearing biological materials, but also provide useful guidelines for designing bioinspired synthetic composites for protective applications.