Integrated sensing and communication (ISAC) is recognized as one of the key enabling technologies for sixth-generation (6G) wireless communication networks, facilitating diverse emerging applications and services in an energy and cost-efficient manner. This paper proposes a multi-user multi-target ISAC system to enable full-space coverage for communication and sensing tasks. The proposed system employs a hybrid simultaneous transmission and reflection reconfigurable intelligent surface (STAR-RIS) comprising active transmissive and passive reflective elements. In the proposed scheme, the passive reflective elements support communication and sensing links for nearby communication users and sensing targets, while low-power active transmissive elements are deployed to improve sensing performance and overcome high path attenuation due to multi-hop transmission for remote targets. Moreover, to optimize the transmissive/reflective coefficients of the hybrid STAR-RIS, a semi-definite relaxation (SDR)-based algorithm is proposed. Furthermore, to evaluate sensing performance, signal-to-interference-noise ratio (SINR) and Cramer-Rao bound (CRB) metrics have been derived and investigated via conducting extensive computer simulations.