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Continuum robots have a broad application prospect in space operations, medical surgery, agricultural harvesting and other fields due to their advantages of good dexterity, flexibility, and human-machine safety. Inspired by biological fishbone structure, a bioinspired fishbone continuum robot was proposed, which had compact structure, light weight, and high flexibility. However, the cross-arranged rigid-flexible-soft coupling multi-section structure of the bioinspired fishbone continuum robot made its statics modeling more difficult. The statics analysis of the bioinspired fishbone continuum robot was carried out based on Cosserat theory considering the coupling of its driving cables and elastic backbone. A Cosserat-rod model for the bioinspired fishbone continuum backbone and a Cosserat-string model for its driving cable were established. Furthermore, the coupling model of the Cosserat-rod and Cosserat-string models was established, which can be used to successfully predict the deformation laws of one bioinspired fishbone unit and two cross-arranged fishbone units under different cable driving forces and different tip loads. Compared with the experimental results, it was proved that the errors of theoretical value were within 1.5mm, which was 1.2% of its length. The research result can provide a configuration for rigid-flexible-soft continuum robots, as well as a theoretical reference for the statics modeling of cabledriven rigid-flexible-soft coupling continuum robots.