Non-line-of-sight (NLOS) imaging of hidden objects is a challenging yet vital task, facilitating important applications such as rescue operations, medical imaging, and autonomous driving. In this paper, we attempt to develop a computational steady-state NLOS localization framework that works accurately and robustly under various illumination conditions. For this purpose, we build a physical NLOS image acquisition hardware system and a corresponding virtual setup to obtain real-captured and simulated steady-state NLOS images under different ambient illuminations. Then, we utilize the captured NLOS images to train/fine-tune a multi-task convolutional neural network (CNN) architecture to perform simultaneous background illumination correction and NLOS object localization. Evaluation results on both stimulated and real-captured NLOS images demonstrate that the proposed method can effectively suppress severe disturbance caused by the variation of ambient light, significantly improving the accuracy and stability of steady-state NLOS localization using consumer-grade RGB cameras. The proposed method potentially paves the way to develop practical steady-state NLOS imaging solutions for around-the-clock and all-weather operations.