A method is introduced to assess the tensile strength of carbon nanofibers, carbon nanotubes (CNTs), and linear chains of carbon atoms (C-chains) obtained from thin amorphous carbon films by electron irradiation. Transmission electron microscopy images show that the nanofibers undergo a radiation-induced necking process, characterized by CNT formation and often followed by the formation of a C-chain. Simulations of the necking process are carried out to determine the tensile stress supported by the nanofiber and CNT neck.