We measure the metallicities of 374 red giant branch (RGB) stars in the isolated, quenched dwarf galaxy Tucana using Hubble Space Telescope narrowband (F395N) calcium H and K imaging. Our sample is a factor of ∼7 larger than what is available from previous studies. Our main findings are as follows. (i) A global metallicity distribution function (MDF) with $\langle {\rm{[Fe/H]}}\rangle =-{1.55}_{-0.04}^{+0.04}$ and ${\sigma }_{{\rm{[Fe/H]}}}={0.54}_{-0.03}^{+0.03}$ . (ii) A metallicity gradient of ‑0.54 ± 0.07 dex ${R}_{e}^{-1}$ (‑2.1 ± 0.3 dex kpc^‑1) over the extent of our imaging (∼2.5 R _e ), which is steeper than literature measurements. Our finding is consistent with predicted gradients from the publicly available FIRE-2 simulations, in which bursty star formation creates stellar population gradients and dark matter cores. (iii) Tucana's bifurcated RGB has distinct metallicities: a blue RGB with $\langle {\rm{[Fe/H]}}\rangle =-{1.78}_{-0.06}^{+0.06}$ and ${\sigma }_{{\rm{[Fe/H]}}}={0.44}_{-0.06}^{+0.07}$ and a red RGB with $\langle {\rm{[Fe/H]}}\rangle =-{1.08}_{-0.07}^{+0.07}$ and ${\sigma }_{{\rm{[Fe/H]}}}={0.42}_{-0.06}^{+0.06}$ . (iv) At fixed stellar mass, Tucana is more metal-rich than Milky Way satellites by ∼0.4 dex, but its blue RGB is chemically comparable to the satellites. Tucana's MDF appears consistent with star-forming isolated dwarfs, though MDFs of the latter are not as well populated. (v) About 2% of Tucana's stars have [Fe/H] < ‑3% and 20% have [Fe/H] > ‑1. We provide a catalog for community spectroscopic follow-up.