  Tachyonic Kaluza-Klein modes normally appear from the compactification of
extra timelike dimensions with scale $L$ and lead to some instabilities of
physical systems. We calculate the contribution of tachyonic gravitons to the
self energies of spherical massive bodies of radius $R$ and discuss their
possible gravitational stability. We find that some spherical bodies can be
stable at critical radii $R=2\pi L p$ for some positive integers $p$. We also
prove the generic property of massive bodies that for the range $0<R\leq\pi L$
the gravitational force due to the ordinary massless graviton exchange is
screened by the Kaluza-Klein mode exchange of tachyonic gravitons.
