We present a broadband (similar to 0.3-70 keV) spectral and temporal analysis of NuSTAR observations of the luminous infrared galaxy NGC 6240 combined with archival Chandra, XMM-Newton, and BeppoSAX data. NGC 6240 is a galaxy in a relatively early merger state with two distinct nuclei separated by similar to 1.''5. Previous Chandra observations resolved the two nuclei and showed that they are both active and obscured by Compton-thick material. Although they cannot be resolved by NuSTAR, we were able to clearly detect, for the first time, both the primary and the reflection continuum components thanks to the unprecedented quality of the NuSTAR data at energies >10 keV. The NuSTAR hard X-ray spectrum is dominated by the primary continuum piercing through an absorbing column density which is mildly optically thick to Compton scattering (tau similar or equal to 1.2, N-H similar to 1.5 x 10(24) cm(-2)). We detect moderately hard X-ray (>10 keV) flux variability up to 20% on short (15-20 ks) timescales. The amplitude of the variability is largest at similar to 30 keV and is likely to originate from the primary continuum of the southern nucleus. Nevertheless, the mean hard X-ray flux on longer timescales (years) is relatively constant. Moreover, the two nuclei remain Compton-thick, although we find evidence of variability in the material along the line of sight with column densities N-H <= 2 x 1023 cm(-2) over long (similar to 3-15 yr) timescales. The observed X-ray emission in the NuSTAR energy range is fully consistent with the sum of the best-fit models of the spatially resolved Chandra spectra of the two nuclei.