  We study some properties of static spherically symmetric elastic bodies in
general relativity using both analytical and numerical tools. The materials
considered belong to the class of John elastic materials and reduce to perfect
fluids when the rigidity parameter is set to zero. We find numerical support
that such elastic bodies exist with different possible shapes (balls, single
shells and multiple shells) and that their gravitational redshift can be very
large ($z\approx 2.8$) without violating the dominant energy condition.
Moreover we show that the elastic body has finite radius even in the case when
the constitutive equation of the elastic material is a perturbation of a
polytropic fluid without finite radius, thereby concluding that such fluids are
structurally unstable within the larger class of elastic matter models under
study.
