Bio-inspired serpentine robots show strong potential for operation in unstructured environments, yet existing systems often lack reliable modular docking, adaptive grasping, and an effective balance between structural stiffness and motion dexterity. This study proposes a Modular Omnidirectional Serpentine Robot (MOSR) that integrates a DNA-inspired tendon-driven helical continuum unit, an underactuated omnidirectional spherical docking gripper, and adaptive gripper fingers within a single module. The helical continuum unit provides two-degree-of-freedom compliant bending while improving axial stiffness through interleaved helices and a central constraint structure. The spherical docking gripper adopts a linkage-spring-slider underactuated mechanism to accommodate effective-diameter variations and support stable one-to-one and one-to-many docking. Gripper kinematics are modeled using an improved Denavit–Hartenberg method, and the workspace is verified by MATLAB simulation. Equivalent torsional and bending stiffness models are established for the helical continuum unit and validated by finite element analysis, with mean relative errors of 11.57% and 17.95%, respectively. Docking-angle analysis based on the receiver polar angle (θ_rec) and engager azimuth angle (θ_eng) shows that 61.1% of the receiver surface lies within the feasible docking region at an opening distance of 5.7 mm. A 3D-printed Polyamide 1010 prototype achieves a locomotion speed of 15.3 mm/s on grass and demonstrates terrain traversal, planar steering, obstacle crossing, adaptive grasping, and stable straight and oblique docking. These results verify the feasibility of integrating locomotion, grasping, and modular reconfiguration within a single serpentine robot module.