Robotic systems are frequently deployed in missions that are dull, dirty, and dangerous, where ensuring their safety is of paramount importance when designing stabilizing controllers to achieve their desired goals. This paper addresses the problem of safe circumnavigation around a hostile target by a nonholonomic robot, with the objective of maintaining a desired safe distance from the target. Our solution approach involves incorporating an auxiliary circle into the problem formulation, which assists in navigating the robot around the target using available range-based measurements. By leveraging the concept of a barrier Lyapunov function, we propose a novel control law that ensures stable circumnavigation around the target while preventing the robot from entering the safety circle. This controller is designed based on a parameter that depends on the radii of three circles, namely the stabilizing circle, the auxiliary circle, and the safety circle. By identifying an appropriate range for this design parameter, we rigorously prove the stability of the desired equilibrium of the closed-loop system. Additionally, we provide an analysis of the robot's motion within the auxiliary circle, which is influenced by a gain parameter in the proposed controller. Simulation and experimental results are presented to illustrate the key theoretical developments.