  We introduce a rigorous and general framework to study systematically
self-gravitating elastic materials within general relativity, and apply it to
investigate the existence and viability, including radial stability, of
spherically symmetric elastic stars. We present the mass-radius ($M-R$) diagram
for various families of models, showing that elasticity contributes to increase
the maximum mass and the compactness up to $\approx 22\%$, thus supporting
compact stars with mass well above two solar masses. Some of these elastic
stars can reach compactness as high as $GM/(c^2R)\approx 0.35$ while remaining
stable under radial perturbations and satisfying all energy conditions and
subluminal wave propagation, thus being physically realizable models of stars
with a light ring. We provide numerical evidence that radial instability occurs
for central densities larger than that corresponding to the maximum mass, as in
the perfect-fluid case. Elasticity may be a key ingredient to build consistent
models of exotic ultracompact objects and black-hole mimickers, and can also be
relevant for a more accurate modelling of the interior of neutron stars.
