Eye-tracking is prevalent in scientific and commercial applications. Recent computer vision and deep learning methods enable eye-tracking with off-the-shelf webcams and reduce dependence on expensive, restrictive hardware. However, such deep learning methods have not yet been applied and evaluated for remote, online psychological experiments. In this study, we tackle important challenges faced in remote eye-tracking setups and systematically evaluate appearance-based deep learning methods of gaze tracking and blink detection. From their own home and laptop, 65 participants performed a battery of eye-tracking tasks requiring different eye movements that characterized gaze and blink prediction accuracy over a comprehensive list of measures. We improve the state-of-the-art for eye-tracking during online experiments with an accuracy of 2.4° and precision of 0.47° which reduces the gap between lab-based and online eye-tracking performance. We release the experiment template, recorded data, and analysis code with the motivation to escalate affordable, accessible, and scalable eye-tracking.