Toward the future goal of creating a lung surgery system featuring multiple tentacle-like robots, we present a new folding concept for continuum robots that enables them to squeeze through openings smaller than the robot's nominal diameter (e.g., the narrow space between adjacent ribs). This is facilitated by making the disks along the robot's backbone foldable. We also demonstrate that such a robot can feature not only straight, but also curved tendon routing paths, thereby achieving a diverse family of conformations. We find that the foldable robot performs comparably, from a kinematic perspective, to an identical non-folding continuum robot at varying deployment lengths. This work paves the way for future applications with a continuum robot that can fold and fit through smaller openings, with the potential to reduce invasiveness during surgical tasks.