  We present deep VLT and HST observations of the nearest examples of
calcium-rich 'gap' transients -- rapidly evolving transient events, with a
luminosity intermediate between novae and supernovae. These sources are
frequently found at large galactocentric offsets, and their progenitors remain
mysterious. Our observations find no convincing underlying quiescent sources
coincident with the locations of these transients, allowing us to rule out a
number of potential progenitor systems. The presence of surviving massive-star
binary companions (or other cluster members) are ruled out, providing an
independent rejection of a massive star origin for these events. Dwarf
satellite galaxies are disfavoured unless one invokes as yet unknown conditions
that would be extremely favourable for their production in the lowest mass
systems. Our limits also probe the majority of the globular cluster luminosity
function, ruling out the presence of an underlying globular cluster population
at high significance, and thus the possibility that they are created via
dynamical interactions in dense globular cluster cores. Given the lack of
underlying systems, previous progenitor suggestions have difficulty reproducing
the remote locations of these transients, even when considering solely
halo-borne progenitors. Our preferred scenario is that calcium-rich transients
are high velocity, kicked systems, exploding at large distances from their
natal site. Coupled with a long-lived progenitor system post-kick, this
naturally explains the lack of association these transients have with their
host stellar light, and the extreme host-offsets exhibited. Neutron star --
white dwarf mergers may be a promising progenitor system in this scenario.
