Simultaneously transmitting and reflecting reconfigurable intelligent surface (STAR-RIS) is becoming a prospective technology in realizing extremely low power transmission and seamless coverage for the sixth generation (6G) wireless communication era. Motivated by this, we study the secrecy performance of the STAR-RIS assisted non-orthogonal multiple access (NOMA) networks. To be more practical, we assume that all nodes suffer from residual hardware impairments (RHIs). To characterize the performance of the considered network, the analytical expressions of the secrecy outage probability (SOP) for NOMA users over the Nakagami-$m$ fading channels are derived. Meanwhile, the asymptotic behaviors of SOP at high signal-to-noise ratio (SNR) are also studied. The numerical results show that: 1) The existence of the RHIs reduces the secrecy performance of the network under consideration. 2) The increase in the number of configurable elements $K$ will improve the system secrecy performance.