We report gapless quantum spin liquid behavior in the layered triangular Sr_{3}CuSb_{2}O_{9} system. X-ray diffraction shows superlattice reflections associated with atomic site ordering into triangular Cu planes well separated by Sb planes. Muon spin relaxation measurements show that the S=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 θ_{CW}≃-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 (C_{m}) below 5 K reveals a C_{m}=γT+αT^{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 ∼36 K using a Dirac spin liquid ansatz which is not far from the values inferred from microscopic density functional theory calculations (∼45 K) as well as high-temperature susceptibility analysis (∼70 K). The linear specific heat coefficient is about 18 mJ/mol K^{2} which is somewhat larger than for typical Fermi liquids.