  We present Very Long Baseline Interferometry observations of a faint and
low-luminosity ($L_{\rm 1.4 GHz} < 10^{27}~\mbox{W Hz}^{-1}$) Gigahertz-Peaked
Spectrum (GPS) and Compact Steep Spectrum (CSS) sample. We select eight sources
from deep radio observations that have radio spectra characteristic of a GPS or
CSS source and an angular size of $\theta \lesssim 2$ arcsec, and detect six of
them with the Australian Long Baseline Array. We determine their linear sizes,
and model their radio spectra using Synchrotron Self Absorption (SSA) and Free
Free Absorption (FFA) models. We derive statistical model ages, based on a
fitted scaling relation, and spectral ages, based on the radio spectrum, which
are generally consistent with the hypothesis that GPS and CSS sources are young
and evolving. We resolve the morphology of one CSS source with a radio
luminosity of $10^{25}~\mbox{W Hz}^{-1}$, and find what appear to be two
hotspots spanning 1.7 kpc. We find that our sources follow the turnover-linear
size relation, and that both homogenous SSA and an inhomogeneous FFA model can
account for the spectra with observable turnovers. All but one of the FFA
models do not require a spectral break to account for the radio spectrum, while
all but one of the alternative SSA and power law models do require a spectral
break to account for the radio spectrum. We conclude that our low-luminosity
sample is similar to brighter samples in terms of their spectral shape,
turnover frequencies, linear sizes, and ages, but cannot test for a difference
in morphology.
