Exceptional points (EPs) are singularities that arise in non-Hermitian physics. Current research efforts focus only on systems supporting isolated EPs characterized by increased sensitivity to external perturbations, which makes them potential candidates for building next generation optical sensors. On the downside, this feature is also the Achilles heel of these devices: they are very sensitive to fabrication errors and experimental uncertainties. To overcome this problem, we introduce a new design concept for implementing photonic EPs that combine the robustness required for practical use together with their hallmark sensitivity. Particularly, our proposed structure exhibits a hypersurface of Jordan EPs embedded in a larger space, and having the following peculiar features: (1) A large class of undesired perturbations shift the operating point along the exceptional surface (ES), thus, leaving the system at another EP which explains the robustness; (2) Perturbations due to back reflection or backscattering force the operating point out of the ES, leading to enhanced sensitivity. Importantly, our proposed geometry is relatively easy to implement using standard photonics components and the design concept can be extended to other physical platforms such as microwave or acoustics.