The aim of this work is to design and model a novel modular bionic soft robot for crawling and crossing obstacles. The modular bionic soft robot is composed of several serial driving soft modules, each module is composed of two parallel soft actuators. By analyzing the influence of working pressure and manufacturing size on the stiffness of the modular bionic soft robot, the nonlinear variable stiffness model of the modular bionic soft robot is established. Based on this model, the spatial states and design parameters of the modular bionic soft robot are discussed when the modular bionic soft robot can pass through the obstacle. Experiments show that when the inflation air pressure of the modular bionic soft robot is 70 kPa, its speed can reach 7.89 mm/s and the height of obstacles passed by it can reach 42.8 mm. The feasibility of the proposed modular bionic soft robot and nonlinear variable stiffness model is verified by locomotion experiments.