Traditional self-interference cancellation (SIC) methods are common in full-duplex (FD) integrated sensing and communication (ISAC) systems. However, exploring new SIC schemes is important due to the limitations of traditional approaches. With the challenging limitations of traditional SIC approaches, this paper proposes a novel simultaneous transmitting and reflecting reconfigurable intelligent surface (STAR-RIS)-enabled FD ISAC system, where STAR-RIS enhances simultaneous communication and target sensing and reduces self-interference (SI) to a level comparable to traditional SIC approaches. The optimization of maximizing the sensing signal-to-interference-plus-noise ratio (SINR) and the communication sum rate, both crucial for improving sensing accuracy and overall communication performance, presents significant challenges due to the non-convex nature of these problems. Therefore, we develop alternating optimization algorithms to iteratively tackle these problems. Specifically, we devise the semi-definite relaxation (SDR)-based algorithm for transmit beamformer design. For the reflecting and refracting coefficients design, we adopt the successive convex approximation (SCA) method and implement the SDR-based algorithm to tackle the quartic and quadratic constraints. Simulation results validate the effectiveness of the proposed algorithms and show that the proposed deployment can achieve better performance than that of the benchmark using the traditional SIC approach without STAR-RIS deployment.