We characterize the ability of the Dark Energy Camera (DECam) to perform relative astrometry across its 500. Mpix, 3-deg(2) science field of view and across four years of operation. This is done using internal comparisons of similar to 4 x10(7) measurements of high signal-to-noise ratio stellar images obtained in repeat visits to fields of moderate stellar density, with the telescope dithered to move the sources around the array. An empirical astrometric model includes terms for optical distortions; stray electric fields in the CCD detectors; chromatic terms in the instrumental and atmospheric optics; shifts in CCD relative positions of up to approximate to 10 mu m when the DECam temperature cycles; and low-order distortions to each exposure from changes in atmospheric refraction and telescope alignment. Errors in this astrometric model are dominated by stochastic variations with typical amplitudes of 10-30. mas (in a 30 s exposure) and 5'-10' coherence length, plausibly attributed to Kolmogorov-spectrum atmospheric turbulence. The size of these atmospheric distortions is not closely related to the seeing. Given an astrometric reference catalog at density approximate to 0.7 arcmin(-2), e.g., from Gaia, the typical atmospheric distortions can be interpolated to approximate to 7 mas rms accuracy (for 30 s exposures) with 1' coherence length in residual errors. Remaining detectable error contributors are 2- 4. mas rms from unmodelled stray electric fields in the devices, and another 2- 4. mas rms from focal plane shifts between camera thermal cycles. Thus the astrometric solution for a single DECam exposure is accurate to 3- 6. mas (approximate to 0.02 pixels, or approximate to 300 nm) on the focal plane, plus the stochastic atmospheric distortion.