This paper presents materials and core/shell architectures that provide optimized mechanical properties in packages for stretchable electronic systems. Detailed experimental and theoretical studies quantitatively connect the geometries and elastic properties of the constituent materials to the overall mechanical responses of the integrated systems, with a focus on interfacial stresses, effective modulus, and maximum extent of elongation. Specific results include core/shell designs that lead to peak values of the shear and normal stresses on the skin that remain less than 10 kPa even for applied strains of up to 20%, thereby inducing minimal somatosensory perception of the device on the human skin. Additional, strain-limiting mesh structures embedded in the shell improve mechanical robustness by protecting the active components from strains that would otherwise exceed the fracture point. Demonstrations in precommercial stretchable electronic systems illustrate the utility of these concepts.