Typical continuum robots, such as pneumatic and tendon-driven robots, have a restricted section length and require a large external component for pulleys and a compressor, making them unsuitable for locomotion. This paper presents a new mobile continuum robot design with virtually unlimited extensible sections. A driving unit, which has a mechanism similar to the rack-and-pinion, consists of three DC motors with gears, each of which moves each flexible tube. The rotation of the motor translates the flexible tube, which has a helical groove on the surface that meshes with the gear. The long flexible tube provides a large traveling distance as long as it does not buckle. The elongation and bending motion of each section may be controlled during operation by varying the speed of each flexible tube. This design not only allows the expansion of the robot to otherwise unreachable work areas but also improves the locomotion velocity by generating a large traveling distance of the flexible tubes. The most important point in this paper is to use multiple driving units for locomotion. Since all the driving units can be mounted on the same tubes, by increasing the number of them, the robot can take various forms without expanding its diameter. A preliminary prototype was built, and its crawling locomotion performance was tested using two operating sequences. The results indicate that earthworm-like locomotion can be achieved with good performance by elongating the sections even when the ground is slippery. The proposed design can be easily be rebuilt by anyone with access to a basic 3D printer.