Recently, a new class of radio transients in the 5 GHz band and with durations of the order of hours to days, lacking any visible-light counterparts, was detected by Bower and collaborators. We present new deep near-infrared (IR) observations of the field containing these transients, and find no counterparts down to a limiting magnitude of K = 20.4 mag. We argue that the bright (>1 Jy) radio transients recently reported by Kida et al. are consistent with being additional examples of the Bower et al. transients. We refer to these groups of events as "long-duration radio transients." The main characteristics of this population are: timescales longer than 30 minutes but shorter than several days; very large rate, ∼10 3 deg −2 yr −1 ; progenitor's sky surface density of >60 deg −2 (at 95% confidence) at Galactic latitude ∼40 • ; 1.4-5 GHz spectral slopes, f ν ∝ ν α , with α 0; and most notably the lack of any X-ray, visible-light, near-IR, and radio counterparts in quiescence. We discuss putative known astrophysical objects that may be related to these transients and rule out an association with many types of objects including supernovae, gamma-ray bursts, quasars, pulsars, and M-dwarf flare stars. Galactic brown dwarfs or some sort of exotic explosions in the intergalactic medium remain plausible (though speculative) options. We argue that an attractive progenitor candidate for these radio transients is the class of Galactic isolated old neutron stars (NSs). We confront this hypothesis with Monte Carlo simulations of the space distribution of old NSs, and find satisfactory agreement for the large areal density. Furthermore, the lack of quiescent counterparts is explained quite naturally. In this framework, we find: the mean distance to events in the Bower et al. sample is of order kpc; the typical distance to the Kida et al. transients are constrained to be between 45 pc and 2 kpc (at the 95% confidence level); these events should repeat with a timescale of order several months; and sub-mJy level bursts should exhibit Galactic latitude dependence. We discuss two possible mechanisms giving rise to the observed radio emission: incoherent synchrotron emission and coherent emission. We speculate that if the latter is correct, the long-duration radio transients are sputtering ancient pulsars or magnetars and will exhibit pulsed emission.