  We report on X-ray and optical observations of the X-ray burster EXO
1747-214. This source is an X-ray transient, and its only known outburst was
observed in 1984-1985 by the EXOSAT satellite. We re-analyzed the EXOSAT data
to derive the source position, column density, and a distance upper limit using
its peak X-ray burst flux. We observed the EXO 1747-214 field in 2003 July with
the Chandra X-ray Observatory to search for the quiescent counterpart. We found
one possible candidate just outside the EXOSAT error circle, but we cannot rule
out the possibility that the source is unrelated to EXO 1747-214. Our
conclusion is that the upper limit on the unabsorbed 0.3-8 keV luminosity is L
< 7E31 erg/s, making EXO 1747-214 one of the faintest neutron star transients
in quiescence. We compare this luminosity upper limit to the quiescent
luminosities of 19 neutron star and 14 black hole systems and discuss the
results in the context of the differences between neutron stars and black
holes. Based on the theory of deep crustal heating by Brown and coworkers, the
luminosity implies an outburst recurrence time of >1300 yr unless some form of
enhanced cooling occurs within the neutron star. The position of the possible
X-ray counterpart is consistent with three blended optical/IR sources with
R-magnitudes between 19.4 and 19.8 and J-magnitudes between 17.2 and 17.6. One
of these sources could be the quiescent optical/IR counterpart of EXO 1747-214.
