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.
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