  Using color-magnitude diagrams from deep archival Hubble Space Telescope
imaging, we self-consistently measure the star formation history of Eridanus II
(Eri II), the lowest-mass galaxy ($M_{\star}(z=0) \sim 10^5 M_{\odot}$) known
to host a globular cluster (GC), and the age, mass, and metallicity of its GC.
The GC ($\sim13.2\pm0.4$ Gyr, $\langle$[Fe/H]$\rangle = -2.75\pm0.2$ dex) and
field (mean age $\sim13.5\pm0.3$ Gyr, $\langle$[Fe/H]$\rangle = -2.6\pm0.15$
dex) have similar ages and metallicities. Both are reionization-era relics that
formed before the peak of cosmic star and GC formation ($z\sim2-4$). The
ancient star formation properties of Eri II are not extreme and appear similar
to $z=0$ dwarf galaxies. We find that the GC was $\lesssim4$ times more massive
at birth than today and was $\sim$10% of the galaxy's stellar mass at birth. At
formation, we estimate that the progenitor of Eri II and its GC had $M_{\rm UV}
\sim -7$ to $-12$, making it one of the most common type of galaxy in the early
Universe, though it is fainter than direct detection limits, absent
gravitational lensing. Archaeological studies of GCs in nearby low-mass
galaxies may be the only way to constrain GC formation in such low-mass
systems. We discuss the strengths and limitations in comparing archaeological
and high redshift studies of cluster formation, including challenges stemming
from the Hubble Tension, which introduces uncertainties into the mapping
between age and redshift.
