Red and Blue shift

You're asking about the red and blue shift of stars and galaxies? Very accurately, since you have spectral absorption lines to compare with a reference.

 

I've done some googling and haven't found too much, but this paper implies an accuracy of 0.1% to 1%.
http://www8.nationalacademies.org/astro2010/DetailFileDisplay.aspx?id=208

Redshift is measured with a spectrograph: http://en.wikipedia.org/wiki/Faint_Object_Spectrograph

Redshift doesn't directly measure speed or distance, it directly measures the shift in a spectrum of light.

 

Those spectrographs are looking at light that was emitted from an object. The spectral lines are absorbed at the SOURCE, not the instrument. In other words, the line at 393 nm absorbed by calcium is a result of actual calcium in the star or galaxy absorbing the light, not calcium here on Earth. The light that is at 392 nm is NOT being absorbed by calcium, the line is wider than 1 nm because of several different effects, such as the rotation of the star. (Part of the star is moving away and part is moving towards us, so the line is wider than it would be otherwise) When we look at the patterns of lines in the spectrum and compare it with our own lines here on Earth we see a difference where the lines are shifted to the red end, aka redshifted. Measuring the difference between the lines we observe from the object and our comparison here in the lab we can tell how fast something is moving away from or towards us.

 

This would imply a velocity of ~1/400c or about 1000km/s, which is equivalent to a rotational period of about an hour for a sun-sized star. As comparison: The sun's surface needs 25-30 days for a rotation (depends on the latitude), and 1/400c is much more than the escape velocity of stars.


There is no theoretic limit on the precision, and the technical limits depend on the instruments, measurement time and brightness of the source. Exoplanet searches can measure the radial velocity of nearby stars with a precision of ~1m/s which is equivalent to a relative precision of ~3*10^(-9). The key point here: While the actual lines are broader than this, the center of those lines can be measured with a precision better than the line widths.