Electrons of dark current (DC), generated in high-gradient superconducting RF cavities (SRF) due to field emission, can be accelerated up to very high energies-19 GeV in the case of the International Linear Collider (ILC) main linac-before they are removed by focusing and steering magnets. Electromagnetic and hadron showers generated by such electrons can represent a significant radiation threat to the linac equipment and personnel. In our study, an operational scenario is analysed which is believed can be considered as the worst case scenario for the main linac regarding the DC contribution to the radiation environment in the main linac tunnel. A detailed modeling is performed for the DC electrons which are emitted from the surface of the SRF cavities and can be repeatedly accelerated in the high-gradient fields in many SRF cavities. Results of MARS15 Monte Carlo calculations, performed for the current main linac tunnel design, reveal that the prompt dose design level of 25 {mu}Sv/hr in the service tunnel can be provided by a 2.3-m thick concrete wall between the main and service tunnels.