  Ba$_3$Mn$_2$O$_8$ is a geometrically frustrated spin dimer compound. We
investigate the effect of site disorder on the zero field phase diagram of this
material by considering the solid solution
Ba$_{3}$(Mn$_{1-x}$V$_{x}$)$_{2}$O$_{8}$, where nonmagnetic V$^{5+}$ ions
partially substitute magnetic Mn$^{5+}$ ions. This substitution results in
unpaired $S=1$ moments for half-substituted dimers, which are ungapped and
therefore susceptible to types of magnetic order not present in the parent
compound. AC susceptibility measurements of compositions between $x=0.046$ and
$x=0.84$ show a sharp frequency- and composition-dependent kink at temperatures
below 210mK, suggesting that unpaired spins form a spin glass. The case for a
glassy state is made clearer by the absence of any sharp features in the
specific heat. However, Ba$_{3}$(Mn$_{1-x}$V$_{x}$)$_{2}$O$_{8}$ is not a
paradigmatic spin glass. Whereas both the freezing temperature and the Weiss
temperature (determined from susceptibility above 1K) vary strongly as a
function of composition, the heat capacity per unpaired spin is found to be
insensitive (above the glass transition) to the density of unpaired spins for
the broad regime $0.18\leq x \leq 0.84$. This surprising result is consistent
with a scenario in which nearest-neighbor unpaired spins form local, possibly
fluctuating, spin-singlets prior to the eventual spin freezing. The spin glass
state is only found for temperatures below the energy scale of single-ion
anisotropy, suggestive this plays a significant role in determining the
eventual ground state. Possible ground states in the "dilute" limit ($x < 0.04$
and $x > 0.9$) are also discussed.
