In this paper, we investigate an active simultaneously transmitting and reflecting reconfigurable intelligent surface (STAR-RIS) assisted integrated sensing and communications (ISAC) system, where the dual-function base station (DFBS) operates in full-duplex (FD) mode to provide communication services and performs targets sensing simultaneously. Meanwhile, we consider multiple targets and multiple users scenario as well as the self-interference at the FD DFBS. Through jointly optimizing the DFBS and active STAR-RIS beamforming under different work modes, our purpose is to achieve the maximum communication sum-rate, while satisfying the minimum radar signal-to-interference-plus-noise ratio (SINR) constraint, the active STAR-RIS hardware constraints and the total power constraint of DFBS and active STAR-RIS. To tackle the complex non-convex optimization problem formulated, an efficient alternating optimization algorithm is proposed. Specifically, the fractional programming method is first leveraged to turn the original problem into a more tractable one, and subsequently the transformed problem is decomposed into several sub-problems. Next, we develop a derivation method to obtain the closed-form expression of the radar receiving beamforming, and then the DFBS transmit beamforming is optimized under the radar SINR requirement and total power constraints. After that, the active STAR-RIS reflection and transmission beamforming are optimized by majorization minimization, complex circle manifold and convex optimization techniques. Finally, the proposed schemes are conducted through numerical simulations to show their benefits and efficiency.