The cable-driven continuum robot (CDCR) is a highly significant soft robot that exhibits a lightweight structure, intrinsic safety properties, and a considerable degree of freedom; therefore, it can work well in confined and complex environments. However, commonly used fiber Bragg grating sensors in CDCR systems are ultra-stiff, extremely low in elongation, and lack an adhesion mechanism; this significantly restricts the movement of the robot and tends to delaminate from it, which makes it unsuitable for integrated systems. In this study, a new strategy is developed to enable CDCR perception via skinlike hydrogel sensors made from ionic conductive polyacrylamide/alginate/ nanoclay polymeric composite hydrogels; it exhibits a fracture strain of 1840% and adheres to a CDCR backbone with an adhesion strength of 6.6 kPa. The sensors are sensitive, stable, and reliable, and they can be manually operated to draw portraits using sensing curves as painted lines. Through these sensors, the CDCR acquires proprioception for sensing movements and exteroception for sensing barriers and traps. The hydrogel sensors are further employed to build a closed-loop control system for regulating the bending of the CDCR. This study establishes effective routes for designing sensors and closed-loop systems that can be applied to soft robots.