  We consider theories in which the Standard Model gauge fields propagate in
extra dimensions whose size is around the electroweak scale. The Standard Model
quarks and leptons may either be localized to a brane or propagate in the bulk.
  This class of theories includes models of Scherk-Schwarz supersymmetry
breaking and universal extra dimensions. We consider the problem of stabilizing
the volume of the extra dimensions. We find that for a large class of
stabilization mechanisms the field which corresponds to fluctuations of the
volume remains light even after stabilization, and has a mass in the $10^{-3}$
eV range. In particular this is the case if stabilization does not involve
dynamics at scales larger than the cutoff of the higher dimensional Standard
Model, and if the effective theory below the compactification scale is four
dimensional. The mass of this field is protected against large radiative
corrections by the general covariance of the higher dimensional theory and by
the weakness of its couplings, which are Planck suppressed. Its couplings to
matter mediate forces whose strength is comparable to that of gravity and which
can give rise to potentially observable deviations from Newton's Law at
sub-millimeter distances. Current experiments investigating short distance
gravity can probe extra dimensions too small to be accessible to current
collider experiments. In particular for a single extra dimension stabilized by
the Casimir energy of the Standard Model fields compactification radii as small
as 5 inverse TeV are accessible to current sub-millimeter gravity experiments.
