  The superconducting TMD 4Hb-TaS$_2$ consists of alternating layers of H and T
structures, which in their bulk form are metallic and Mott-insulating,
respectively. Recently, this compound has been proposed as a candidate chiral
superconductor, due to an observed enhancement of the muon spin relaxation at
$T_c$. 4Hb-TaS$_2$ also exhibits a puzzling $T$-linear specific heat at low
temperatures, which is unlikely to be caused by disorder. Elucidating the
origin of this behavior is an essential step in discerning the true nature of
the superconducting ground state. Here, we propose a simple model that
attributes the $T$-linear specific heat to the emergence of a robust multi-band
gapless superconducting state. We show that an extended regime of gapless
superconductivity naturally appears when the pair-breaking scattering rate on
distinct Fermi-surface pockets differs significantly, and the pairing
interaction is predominantly intra-pocket. Using a tight-binding model derived
from first-principle calculations, we show that the pair-breaking scattering
rate promoted by slow magnetic fluctuations on the T layers, which arise from
proximity to a Mott transition, can be significantly different in the various
H-layer dominated Fermi pockets depending on their hybridization with T-layer
states. Thus, our results suggest that the ground state of 4Hb-TaS$_2$ consists
of Fermi pockets displaying gapless superconductivity, which are shunted by
superconducting Fermi pockets that are nearly decoupled from the T-layers.
