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 Motts 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.
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