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We present very long baseline interferometry observations of a faint and low-luminosity (L1.4GHz < 1027 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 θ ≲ 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 1025WHz-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 homogeneous 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. [ABSTRACT FROM AUTHOR]