  In this work we examine the effective four-dimensional world that emanates
from a general class of static spherical Ricci-flat solutions in Kaluza-Klein
gravity in $D$-dimensions. By means of dimensional reduction we obtain a family
of asymptotically flat Schwarzschild-like metrics for which all the components
of the Ricci tensor, except for $R_{11}$, are zero. Although the reduced
spacetime is not empty, it is similar to vacuum in the sense that the effective
matter satisfies an equation of state which is the generalization of $(\rho +
3p) = 0$ for "nongravitating matter" in 4D. In Kaluza-Klein gravity these
Schwarzschild-like metrics describe the exterior of a spherical star without
rotation. In this framework, we generalize the well-known Buchdahl's theorem
for perfect fluid spheres whose mass density does not increase outward. Without
any additional assumptions, we develop the most general expression for the
compactness limit of a star. We provide some numerical values for it, which in
principle are observationally testable and allow us to compare and contrast
different theories and exteriors. We find that in Kaluza-Klein gravity the
compactness limit of a star can be larger than 1/2, without being a black hole:
the general-relativistic upper limit $M/R < 4/9$ is increased as we go away
from the Schwarzschild vacuum exterior. We show how this limit depends on the
number of dimensions of spacetime, and demonstrate that the effects of gravity
are stronger in 4D than in any other number of dimensions.
