Sandwich panels, characterized by their solid exterior and thick, soft core, find extensive applications ranging from maritime to aerospace sectors due to their exceptional resilience and energy absorption/dissipation capabilities. The natural limpet, known for its resilience to mechanical loading, serves as inspiration in the present research to introduce innovative repairable core shells. Limpet-inspired shells with elasto-plastic and brittle behaviors are designed and 3D/4D printed using polylactic acid (PLA) filaments and resins via fused filament fabrication (FFF) and liquid crystal display (LCD) techniques. While PLA filament exhibits an elasto-plastic response with a shape-recovery feature, the resin results in brittle structures for LCD-printed design. The study demonstrates that the FFF-printed PLA shell can fully recover residual plastic deformations, while the LCD-printed PLA shell exhibits a mechanical fracture behavior very similar to the natural limpet with brittle properties. A nonlinear finite element model (FEM) is also developed to replicate the large deformations of the samples under quasi-static compression with a high level of accuracy. Experimental and numerical results reveal that the samples with material, mechanical behavior, and geometry close to the natural limpet result in the maximum energy dissipation per unit mass. Two bio-inspired cores are then tessellated to introduce a new class of sustainable sandwich panels with supreme recoverability, resiliency, and repairability. A three-point bending test is numerically carried out on sandwich panels using FEM, and their energy absorption and dissipation capacities are studied. Results demonstrate that the proposed sandwich panels can achieve almost 7.33 and 1.17 times higher energy dissipation per unit mass than those developed based on a recycled thermoplastic bottle cap core. This pioneering research sets a new benchmark for structural design in terms of resiliency, recoverability, repairability, and sustainability.