We present Nuclear Spectroscopic Telescope Array (NuSTAR) 3-40 keV observations of the optically selected Type 2 quasar (QSO2) SDSS J1034+6001 or Mrk 34. The high-quality hard X-ray spectrum and archival XMM-Newton data can be fitted self-consistently with a reflection-dominated continuum and a strong Fe K alpha fluorescence line with equivalent width > 1 keV. Prior X-ray spectral fitting below 10 keV showed the source to be consistent with being obscured by Compton-thin column densities of gas along the line of sight, despite evidence for much higher columns from multiwavelength data. NuSTAR now enables a direct measurement of this column and shows that N-H lies in the Compton-thick (CT) regime. The new data also show a high intrinsic 2-10 keV luminosity of L2-10 similar to 10(44) erg s (1), in contrast to previous low-energy X-ray measurements where L2-10 less than or similar to 10(43) erg s (1) (i.e., X-ray selection below 10 keV does not pick up this source as an intrinsically luminous obscured quasar). Both the obscuring column and the intrinsic power are about an order of magnitude (or more) larger than inferred from pre-NuSTAR X-ray spectral fitting. Mrk 34 is thus a "gold standard" CT QSO2 and is the nearest non-merging system in this class, in contrast to the other local CT quasar NGC 6240, which is currently undergoing a major merger coupled with strong star formation. For typical X-ray bolometric correction factors, the accretion luminosity of Mrk 34 is high enough to potentially power the total infrared luminosity. X-ray spectral fitting also shows that thermal emission related to star formation is unlikely to drive the observed bright soft component below similar to 3 keV, favoring photoionization instead.