We present the 3D real-space clustering power spectrum of a sample of ∼600 000 luminous red galaxies measured by the Sloan Digital Sky Survey, using photometric redshifts. These galaxies are old, elliptical systems with strong 4000-Å breaks, and have accurate photometric redshifts with an average error of Δz= 0.03. This sample of galaxies ranges from redshift z= 0.2 to 0.6 over 3528 deg2 of the sky, probing a volume of 1.5 h−3 Gpc3, making it the largest volume ever used for galaxy clustering measurements. We measure the angular clustering power spectrum in eight redshift slices and use well-calibrated redshift distributions to combine these into a high-precision 3D real-space power spectrum from k= 0.005 to k= 1 h Mpc−1. We detect power on gigaparsec scales, beyond the turnover in the matter power spectrum, at a ∼2σ significance for k < 0.01 h Mpc−1, increasing to 5.5σ for k < 0.02 h Mpc−1. This detection of power is on scales significantly larger than those accessible to current spectroscopic redshift surveys. We also find evidence for baryonic oscillations, both in the power spectrum, as well as in fits to the baryon density, at a 2.5 σ confidence level. The large volume and resulting small statistical errors on the power spectrum allow us to constrain both the amplitude and the scale dependence of the galaxy bias in cosmological fits. The statistical power of these data to constrain cosmology is ∼1.7 times better than previous clustering analyses. Varying the matter density and baryon fraction, we find ΩM= 0.30 ± 0.03, and Ωb/ΩM= 0.18 ± 0.04, for a fixed Hubble constant of 70 km s−1 Mpc−1 and a scale-invariant spectrum of initial perturbations. The detection of baryonic oscillations also allows us to measure the comoving distance to z= 0.5; we find a best-fitting distance of 1.73 ± 0.12 Gpc, corresponding to a 6.5 per cent error on the distance. These results demonstrate the ability to make precise clustering measurements with photometric surveys.