We analyze the X-ray properties of a near-infrared selected (Ks < 20) sample of spectroscopically identified Extremely Red Objects (R-Ks>5) in a region of the Chandra Deep Field-South using the public 1 Ms observation. The 21 objects were classified, on the basis of deep VLT spectra, in two categories: 13 dusty star-forming galaxies showing [OII] emission, and 8 early-type galaxies with absorption features in their optical spectra. Only one emission line object has been individually detected; its very hard X-ray spectrum and the high intrinsic X-ray luminosity unambiguously reveal the presence of an obscured AGN. Stacking analysis of the remainder 12 emission line objects shows a significant detection with an average luminosity L(X)~8x10^{40} erg/s in the rest-frame 2-10 keV band. The stacked counts of the 8 passive galaxies do not provide a positive detection. We briefly discuss the implications of the present results for the estimate of the Star Formation Rate (SFR) in emission line EROs. Comment: 4 pages, 1 figure. Accepted for publication in ApJ Letters