  Ca$_{3}$Ru$_{2}$O$_{7}$ is a polar metal that belongs to the class of
multiferroic magnetic materials. Here, tiny amounts of Fe doping in the Ru
sites bring about dramatic changes in the electronic and magnetic properties
and generate a complex H-T phase diagram. To date, not much is known about the
ground state of such a system in the absence of magnetic field. By performing
muon-spin spectroscopy (${\mu}$SR) measurements in 5% Fe-doped
Ca$_{3}$Ru$_{2}$O$_{7}$ single crystals, we investigate its electronic
properties at a local level. Transverse-field ${\mu}$SR results indicate a very
sharp normal-to-antiferromagnetic transition at T$_{N}$ = 79.7(1) K, with a
width of only 1 K. Zero-field ${\mu}$SR measurements in the magnetically
ordered state allow us to determine the local fields B$_{i}$ at the muon
implantation sites. By symmetry, muons stopping close to the RuO$_{2}$ planes
detect only the weak nuclear dipolar fields, while those stopping next to
apical oxygens sense magnetic fields as high as 150 mT. In remarkable agreement
with the nominal Fe-doping, a $\sim$ 6% minority of the these muons feel
slightly lower fields, reflecting a local magnetic frustration induced by iron
ions. Finally, B$_{i}$ shows no significant changes across the
metal-to-insulator transition, close to 40 K. We ascribe this surprising lack
of sensitivity to the presence of crystal twinning.

Strongly Correlated Electrons;Materials Science