This letter describes the manufacturing and experimental characterization of novel stretchable strain sensors for continuum robots. The overarching goal of this research is to provide a new solution for the shape sensing of these devices. The sensors are fabricated via direct ink writing, an extrusion-based additive manufacturing technique. Electrically conductive material (i.e., the \textit{ink}) is printed into traces whose electrical resistance varies in response to mechanical deformation. The principle of operation of stretchable strain sensors is analogous to that of conventional strain gauges, but with a significantly larger operational window thanks to their ability to withstand larger strain. Among the different conductive materials considered for this study, we opted to fabricate the sensors with a high-viscosity eutectic Gallium-Indium ink, which in initial testing exhibited high linearity ($R^2 \approx$ 0.99), gauge factor $\approx$ 1, and negligible drift. Benefits of the proposed sensors include (i) ease of fabrication, as they can be conveniently printed in a matter of minutes; (ii) ease of installation, as they can simply be glued to the outside body of a robot; (iii) ease of miniaturization, which enables integration into millimiter-sized continuum robots.