Soft pneumatic gripping strategies are often based on pressurized actuation of structures made of soft elastomeric materials, which limits designs in terms of size, weight, achievable forces, and ease of fabrication. In contrast, fabric‐based inflatable structures offer high stiffness‐to‐weight ratio solutions for soft robotics, but their actuation has been little explored. Herein, a new class of pneumatic soft grippers is presented that exploits the in‐plane overcurvature effect of inextensible fabric flat balloons upon inflation. A star‐shaped gripper contracts radially under pressure producing a gripping force on the object whose intensity can be modulated by the pressure input. First, the kinematics and mechanics of a single V‐shaped actuator are studied through experiments, finite element simulations, and analytical models. Then, these results are leveraged to predict the mechanical response of the entire star, optimize its geometry, and maximize contraction and stiffness. It is shown that the gripping performance can be improved by stacking several stars with silicon‐coated corners. It is expected that the flexibility, robustness, scalability, and ease of fabrication of this methodology will lead to a new generation of lighter and larger actuators capable of developing higher forces and moving delicate and irregularly shaped objects while maintaining reasonable complexity.