The prior works on near-field target localization have mostly assumed ideal hardware models and thus suffer from two limitations in practice. First, extremely large-scale arrays (XL-arrays) usually face a variety of hardware impairments (HIs) that may introduce unknown phase and/or amplitude errors. Second, the existing block coordinate descent (BCD)-based methods for joint estimation of the HI indicator, channel gain, angle, and range may induce considerable target localization error when the target is very close to the XL-array. To address these issues, we propose in this paper a new three-phase HI-aware near-field localization method, by efficiently detecting faulty antennas and estimating the positions of targets. Specifically, we first determine faulty antennas by using compressed sensing (CS) methods and improve detection accuracy based on coarse target localization. Then, a dedicated phase calibration method is designed to correct phase errors induced by detected faulty antennas. Subsequently, an efficient near-field localization method is devised to accurately estimate the positions of targets based on the full XL-array with phase calibration. Additionally, we resort to the misspecified Cramer-Rao bound (MCRB) to quantify the performance loss caused by HIs. Last, numerical results demonstrate that our proposed method significantly reduces the localization errors as compared to various benchmark schemes, especially for the case with a short target range and/or a high fault probability.