The paper presents the data on radiation protection properties of a composite material consisting of the glass-crystalline matrix and nanotubular chrysotile modified by inserting PbWO4 into its structure, as well as the data on key physico-mechanical characteristics of the composite, such as density, ultimate compression strength, microhardness, porosity, water absorption, temperature stability, and thermostability. It was established that in addition to radiation protection properties, the examined material has enhanced practical design characteristics and can be used as a construction material. The propagation of gamma photons with different energy levels through the composite material is examined. A graph is built for dependence of the linear gamma radiation attenuation coefficient (μ) on energy in the range 0.25 < E < 1.4 MeV. The contribution of the Compton effect and the photoeffect into the total linear gamma photon flow attenuation coefficient are considered. It is established that at energy levels from 0.25 to 0.7 MeV, photoeffect makes the largest contribution to the total linear gamma radiation attenuation coefficient, while at energy levels from 0.7 to 1.4 MeV the largest contribution is made by the Compton effect. Error of the linear gamma radiation attenuation coefficient based on estimates and experimental data is very small and equals around 2%, which confirms that the developed model is correct. It is established that the composite possesses enhanced radiation protection characteristics, far exceeding those of iron and slightly (by 10.4%) yielding to pure lead.