  Bi$_2$O$_2$Se is a promising material for next-generation semiconducting
electronics. It exhibits premature metallicity on the introduction of a tiny
amount of electrons, the physics behind which remains elusive. Here we report
on transport and dielectric measurements in Bi$_2$O$_2$Se single crystals at
various carrier densities. The temperature-dependent resistivity ($\rho$)
indicates a smooth evolution from the semiconducting to the metallic state. The
critical concentration for the metal-insulator transition (MIT) to occur is
extraordinarily low ($n_\textrm{c}\sim10^{16}$ cm$^{-3}$). The relative
permittivity of the insulating sample is huge
($\epsilon_\textrm{r}\approx155(10)$) and varies slowly with temperature.
Combined with the light effective mass, a long effective Bohr radius
($a_\textrm{B}^*\approx36(2)$ $\textrm{nm}$) is derived, which provides a
reasonable interpretation of the metallic prematurity according to Mott's
criterion for MITs. The high electron mobility ($\mu$) at low temperatures may
result from the screening of ionized scattering centers due to the huge
$\epsilon_\textrm{r}$. Our findings shed light on the electron dynamics in two
dimensional (2D) Bi$_2$O$_2$Se devices.
