Zinc oxide nanoparticles were synthesized using a plasma discharge method under the effect of ultrasonic cavitation, and films of a polymer composite material were obtained and studied. Comparatively, the mixture of zinc oxide nanoparticles with planar graphene oxide nanostructures were also used as a filler. Using melt compounding technology, composite materials were obtained in the form of thin films consisting of the copolymer of ethylene and vinyl acetate and nanoparticles of zinc oxide and graphene oxide. The obtained samples of composite materials were examined by X-ray phase analysis, X-ray fluorescence analysis, and scanning electron microscopy. The physical, mechanical, and gas transport properties of composite films made of polymers filled with nanoparticles were studied: permeability measurements were carried out for carbon dioxide and helium gases.