Different from conventional reconfigurable intelligent surfaces (RIS), a recent innovation called simultaneous transmitting and reflecting reconfigurable intelligent surface (STAR-RIS) has emerged, aimed at achieving complete 360-degree coverage in communication networks. Additionally, fullduplex (FD) technology is recognized as a potent approach for enhancing spectral efficiency by enabling simultaneous transmission and reception within the same time and frequency resources. In this study, we investigate the performance of a STAR-RIS-assisted FD communication system. The STAR-RIS is strategically placed at the cell-edge to facilitate communication for users located in this challenging region, while cell-center users can communicate directly with the FD base station (BS). We employ a non-orthogonal multiple access (NOMA) pairing scheme and account for system impairments, such as self-interference at the BS and imperfect successive interference cancellation (SIC). We derive closed-form expressions for the ergodic rates in both the up-link and down-link communications and extend our analysis to bidirectional communication between cell-center and cell-edge users. Furthermore, we formulate an optimization problem aimed at maximizing the ergodic sum-rate. This optimization involves adjusting the amplitudes and phase-shifts of the STAR-RIS elements and allocating total transmit power efficiently. To gain deeper insights into the achievable rates of STAR-RIS-aided FD systems, we explore the impact of various system parameters through numerical results.