Rigid-flexible-soft coupled robots are an important development direction of robotics, which face many theoretical and technical challenges in design, manufacturing, and modeling. Inspired by fishbones, we propose a novel cable-driven single-backbone continuum robot which has compact structure, light weight, and high dexterity. Different from the existing single-backbone continuum robot, the middle backbone of the continuum robot is serially formed by multiple cross-arranged bioinspired fishbone units. The proposed bioinspired fishbone unit having good one-dimensional bending properties is a special rigid-flexible-soft structure mainly made by multi-material 3D printing technology. The unique design and manufacturing of the middle backbone bring the continuum robot excellent constant curvature characteristics and reduce the coupling between different motion dimensions, laying a foundation for the continuum robot to have a more accurate theoretical model as well as regular and controllable deformation. Moreover, we build the forward and inverse kinematics model based on the geometric analysis method, and analyze its workspace. Further, the comparison between the experimental and theoretical results shows that the prediction errors of the kinematics model are within desired 0.5 mm. Also, we establish the relation between the cable driving force of the bioinspired fishbone unit and its bending angle which can provide a guidance for the optimization of the continuum robot in the future. The application demos prove that the continuum robot has good dexterity and compliance, and can perform tasks such as obstacle crossing locomotion and narrow space transportation. This work provides new ideas for the bioinspired design and high-precision modeling of continuum robots.