Continuum robots offer many advantages for use in miniature diagnostic and interventional surgical devices. However, the creation of snake-like devices with extremely high slenderness ratios, those with great length and small diameter, remains challenging from a device design perspective. To facilitate improved slenderness ratios, high total accumulated bending angles, and high stiffness and mechanical stability of the active section, we propose the use of a flexible screw-driven mechanism to generate large distal actuation forces while still locating motors and other bulky system components at the base of the device. In comparison to tendon-based designs and push-pull rods, the design avoids the capstan-like buildup of friction. In this work, we present design, fabrication, kinestatic modeling, and experimental validation for a two-degree-of freedom, screw-based, multi-backbone continuum robot. The model that compensates the friction in the system most accurately predicts the behavior of the robot and eliminates most of the hysteresis in the input-output behavior.