We present a method for measuring the Hubble parameter, H(z), and angular diameter distance, D_A(z), from the two-dimensional two-point correlation function, and validate it using LasDamas mock galaxy catalogs. Applying our method to the sample of luminous red galaxies (LRGs) from the Sloan Digital Sky Survey (SDSS) Data Release 7 (DR7), we measure H(z=0.35)=82.1_{-4.9}^{+4.8} km s^{-1}Mpc^{-1}, D_A(z=0.35)=1048_{-58}^{+60} Mpc without assuming a dark energy model or a flat Universe. We find that the derived measurements of H(0.35)r_s(z_d)/c and D_A(0.35)/r_s(z_d) (where r_s(z_d) is the sound horizon at the drag epoch) are nearly uncorrelated, have tighter constraints and are more robust with respect to possible systematic effects. Our galaxy clustering measurements of {H(0.35)r_s(z_d)/c, D_A(0.35)/r_s(z_d)}={0.0434± 0.0018,6.60± 0.26} (with the correlation coefficient r = 0.0604) can be used to combine with cosmic microwave background and any other cosmological data sets to constrain dark energy. Our results represent the first measurements of H(z) and D_A(z) (or H(z)r_s(z_d)/c and D_A(0.35)/r_s(z_d)) from galaxy clustering data. Our work has significant implications for future surveys in establishing the feasibility of measuring both H(z) and D_A(z) from galaxy clustering data. ; Comment: 12 pages, 8 figures. This is the version accepted for publication in MNRAS