We have measured the distance to over 8000 stars: Parallax.
The development of large refractors came to an abrupt end with Yerkes [site of the 40" refractor telescope - still the world's largest]. But not all the telescopes of the past have been relegated to serving as museum pieces. Some are still used for important research work where image quality is especially important. In studies of the cosmic distance scale, traditional measurements of the distances to stars are unreliable beyond 100 light years. Trigonometric parallax is the most precise method of measurement, and the rest of the distance scale is built off it. At this scale, angles are very difficult to measure, so the Allegheny Observatory 30-inch Thaw refractor has been fitted with equipment for multichannel astrometric photometry in order to measure parallax to an unprecedented accuracy of 1 milliarcsecond.
The Yerkes 40-inch has done spectroscopic work and photometry, which is the direct measurement of photons to create intensity traces that are used to measure star magnitudes precisely to 0.01 magnitude. The 40-inch has also re-photographed stars that were originally photographed around 1910 to see how much they have moved in the intervening years. By using the same telescope over a long span of years, the image scale remains the same so that the positions of stars can be measured with significantly higher precision.
Emphasis added.
A good summary on the advancement of precision in measuring the distance to stars over time is [link|http://msc.caltech.edu/workshop/2005/presentations/McAlister.pdf|here] (47 page .pdf). See slides 9 and 10 for the basics. The distance falls out of geometry (to get the angle) and Newton's laws (to get the diameter of Earth's orbit). Over 8000 stars have had their parallax measured (as of 1995).
Cheers,
Scott.