Quasars may have played a key role in limiting the stellar mass of massive galaxies. Identifying those quasars in the process of removing star formation fuel from their hosts is an exciting ongoing challenge in extragalactic astronomy. In this paper, we present X-ray observations of 11 extremely red quasars (ERQs) with L bol ~ 1047 erg s−1 at z = 1.5–3.2 with evidence for high-velocity (v $\geqslant $ 1000 km s−1) [O iii] λ5007 outflows. X-rays allow us to directly probe circumnuclear obscuration and to measure the instantaneous accretion luminosity. We detect 10 out of 11 ERQs available in targeted and archival data. Using a combination of X-ray spectral fitting and hardness ratios, we find that all of the ERQs show signs of absorption in the X-rays with inferred column densities of N H ≈ 1023 cm−2, including four Compton-thick candidates (N H $\geqslant $ 1024 cm−2). We stack the X-ray emission of the seven weakly detected sources, measuring an average column density of N H ~ 8 × 1023 cm−2. The absorption-corrected (intrinsic) 2–10 keV X-ray luminosity of the stack is 2.7 × 1045 erg s−1, consistent with X-ray luminosities of type 1 quasars of the same infrared luminosity. Thus, we find that ERQs are a highly obscured, borderline Compton-thick population, and based on optical and infrared data we suggest that these objects are partially hidden by their own equatorial outflows. However, unlike some quasars with known outflows, ERQs do not appear to be intrinsically underluminous in X-rays for their bolometric luminosity. Our observations indicate that low X-rays are not necessary to enable some types of radiatively driven winds.