The peculiarities of spin-polarized electronic structure of multilayer graphene nanofilm (4-GNL:H) within the framework of Kohn --- Sham approximation were studied in the present work. The calculated band structure and spin-resolved electronic energy spectrum of the 4-GNL:H system were correlated with experimental UPS and XANES spectra of thin hydrogenated a-C:H films. As the band structure calculations show there is a dimensional quantization of energy spectrum in the 4-GNL:H system, and the energy gap of 0.11 eV appears in the spectrum. The self-consistent calculations also predict the existence of itinerant magnetism in the system, conditioned by hydrogen chemisorption.