  We report gapless quantum spin liquid behavior in the layered triangular
Sr$_{3}$CuSb$_{2}$O$_{9}$ (SCSO) system. X-ray diffraction shows superlattice
reflections associated with atomic site ordering into triangular Cu planes
well-separated by Sb planes. Muon spin relaxation ($\mu$SR) measurements show
that the $S = \frac{1}{2}$ moments at the magnetically active Cu sites remain
dynamic down to 65 mK in spite of a large antiferromagnetic exchange scale
evidenced by a large Curie-Weiss temperature $\theta_{\mathrm{cw}} \simeq $
-143 K as extracted from the bulk susceptibility. Specific heat measurements
also show no sign of long-range order down to 0.35 K. The magnetic specific
heat ($\mathit{C}$$_{\mathrm{m}}$) below 5 K reveals a
$\mathit{C}$$_{\mathrm{m}}$ $=$ $\gamma T$ + $\alpha T$$^{2}$ behavior. The
significant $T$$^{2}$ contribution to the magnetic specific heat invites a
phenomenology in terms of the so-called Dirac spinon excitations with a linear
dispersion. From the low-$T$ specific heat data, we estimate the dominant
exchange scale to be $\sim $ 36 K using a Dirac spin liquid ansatz which is not
far from the values inferred from microscopic density functional theory
calculations ($\sim $ 45 K) as well as high-temperature susceptibility analysis
($\sim$ 70 K). The linear specific heat coefficient is about 18 mJ/mol-K$^2$
which is somewhat larger than for typical Fermi liquids.
