We measure the large-scale real-space power spectrum P(k) by using asample of 205,443 galaxies from the Sloan Digital Sky Survey, covering2417 effective square degrees with mean redshift z~0.1. We employ amatrix-based method using pseudo-Karhunen-Loève eigenmodes,producing uncorrelated minimum-variance measurements in 22 k-bands ofboth the clustering power and its anisotropy due to redshift-spacedistortions, with narrow and well-behaved window functions in the range0.02hMpc-1-1. We pay particularattention to modeling, quantifying, and correcting for potentialsystematic errors, nonlinear redshift distortions, and the artificialred-tilt caused by luminosity-dependent bias. Our results are robust toomitting angular and radial density fluctuations and are consistentbetween different parts of the sky. Our final result is a measurement ofthe real-space matter power spectrum P(k) up to an unknown overallmultiplicative bias factor. Our calculations suggest that this biasfactor is independent of scale to better than a few percent fork<0.1hMpc-1, thereby making our results useful forprecision measurements of cosmological parameters in conjunction withdata from other experiments such as the Wilkinson Microwave AnisotropyProbe satellite. The power spectrum is not well-characterized by asingle power law but unambiguously shows curvature. As a simplecharacterization of the data, our measurements are well fitted by a flatscale-invariant adiabatic cosmological model withhΩm=0.213+/-0.023 and σ8=0.89+/-0.02for L* galaxies, when fixing the baryon fractionΩb/Ωm=0.17 and the Hubble parameterh=0.72; cosmological interpretation is given in a companion paper.