  We consider stability properties of spherically symmetric spacetimes of stars
in metric f(R) gravity. We stress that these not only depend on the particular
model, but also on the specific physical configuration. Typically
configurations giving the desired $\gamma_{\rm PPN} \approx 1$ are strongly
constrained, while those corresponding to $\gamma_{\rm PPN} \approx 1/2$ are
less affected. Furthermore, even when the former are found strictly stable in
time, the domain of acceptable static spherical solutions typically shrinks to
a point in the phase space. Unless a physical reason to prefer such a
particular configuration can be found, this poses a naturalness problem for the
currently known metric f(R) models for accelerating expansion of the Universe.
