This is a very challenging problem, that even as of today, is still very challenging. For the closest few hundred stars (and the focus on the discussion is on those close stars exclusively), the methodology to measure the distance is based on parallax measurements. You measure the angle to the star using 2 other reference points (the sun being one of them), and six months later, when you have traveled 300 million kilometers in the sky and are just opposite to the sun, you measure the angle again. Based on this triangulation, you can then estimate the distance to the star in question. See illustration below, and you will immediately understand the mechanism.
Read statistical solutions to this problem, based on good design of experiment, at http://www.analyticbridge.com/forum/topics/can-we-use-data-science-to-measure-distances-to-stars
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