The flexible continuum robotic arm, with its advantages of flexibility, adaptability and smoothness, has significant potential in fields such as healthcare, industry and emergency response. However, the nonlinear kinematic characteristics of the flexible continuum and the problem of end-point error compensation still restrict its application in scenarios with high precision requirements. To address these bottlenecks, inspired by the double helix structure of DNA, a flexible continuum module was designed. Subsequently, considering the stiffness performance of the single module, a rope-driven variable stiffness flexible continuum robotic arm was designed. First, three variable-stiffness modules were constructed via topology-optimized simulation by varying the number of spiral turns. A single-joint three-space transformation model was then established using geometric projection and coordinate transformation, and a multi-joint decoupling algorithm was derived, with bending performance verified by ANSYS simulation. Finally, an experimental platform was built, and bending tests showed maximum angles of 65°, 70°, and 75° for single-module configurations and 68° for the multi-module variable-stiffness configuration. An approximately linear relationship between cable displacement and deformation is established. This study provides key theoretical and technical support for precise applications such as minimally invasive surgery and underwater pipeline inspection.