  Scalar field dark matter likely is able to solve all small-scale cosmology
problems facing the cold dark matter (CDM), and has become an emerging
contender to challenge the CDM. It however requires a particle mass $\sim 1 - 2
\times10^{-22}$eV. We find such an extremely small particle mass can naturally
arise from a non-QCD axion mechanism, under fairly general assumptions that a
few species of self-interacting light particles of comparable masses and a
massless gauge boson decouple from the bright sector since the photon
temperature exceeds 200 GeV, and the axion is the dominant dark matter. These
assumptions also set the axion decay constant scale to several $\times 10^{16}$
GeV. Given the above axion mass range, we further pin down the dark-sector
particles to consist of only one species of fermion and anti-fermion, likely
right-handed neutrinos. With a mass around $92-128$ eV, the dark-sector
particles may constitute a minority population of dark matter. If the gauge
boson lives on SU(2), a dilute instanton gas can contribute about $2.5\%$ of
the total relativistic relics in the cosmic microwave background radiation.
