  The fast radio burst FRB 121102 has repeated multiple times, enabling the
identification of its host galaxy and of a spatially-coincident, compact,
steady (`quiescent') radio synchrotron source. It was proposed that FRB 121102
is powered by a young flaring magnetar, embedded within a decades-old supernova
remnant. Using a time-dependent one-zone model, we show that a single expanding
magnetized electron-ion nebula (powered by the same outbursts likely
responsible for the FRBs) can explain all the basic properties of the quiescent
source (size, flux, self-absorption constraints) and the large but decreasing
rotation measure (RM) of the bursts. The quiescent emission is powered by
relativistic thermal electrons heated at the termination shock of the magnetar
wind, while the RM originates from non-relativistic electrons injected earlier
in the nebula's evolution and cooled through expansion and radiative losses.
The model contains few free parameters, which are tightly constrained by
observations: the total energy injected into the nebula over its history, $\sim
10^{50}-10^{51}$ erg, agrees with the magnetic energy of a millisecond
magnetar; the baryon loading of the magnetar outflow (driven by intermittent
flares) is close to the neutron star escape speed; the predicted source age
$\sim 10-40$ years is consistent with other constraints on the nebula size. For
an energy input rate $\dot{E} \propto t^{-\alpha}$ following the onset of
magnetar activity, we predict secular decay of the RM and quiescent source
flux, which approximately follow ${\rm RM} \propto t^{-(6+\alpha)/2}$ and
$F_{\nu} \propto t^{-(\alpha^2+7\alpha-2)/4}$, respectively.
