A few years ago I wrote, along with a collaborator, a guide to uncertainty analysis (commonly and misleadingly referred to as error analysis) in university physics. Yesterday I had a quick look at this, to see if I should update anything for our new bunch of students. As part of this, I had a look at the list of fundamental constants. I was struck (not for the first time) by the wide difference in their uncertainties.
First Example. The Rydberg constant relates the spacing of spectral lines due to electronic transitions in Hydrogen. Its 'accepted' value (from the CODATA committee) is 10973731.568525 per metre, with an uncertainty in the last digits of 73. That means, the committee is reasonably confident the true value lies between 10973731.568452 and 10973.568598 per metre. That's a staggeringly small uncertainty, about one part in ten to the power 11. (One part in a hundred billion).
Second Example. Newton's constant of gravitation (Big 'G"). This constant describes the gravitational attraction between two masses. It is very difficult to measure in the lab (still, we get our second year students to have a go) because its effect is easily swamped by other things. Normally, of course, we never take any notice of the fact that our cup of coffee on the desk is attracted by gravity to the textbook next to it - the effect of them both being attracted to the earth below, which is so much more massive, swamps this. But the attraction between two nearby objects CAN be measured.
The CODATA value for G is 6.6742 times ten to the power of minus 11 metres cubed per kilogram per second squared, with an uncertainty of 10 in the last digits. That is, its value most probably lies between 6.6732 and 6.6752 times ten to the power of minus 11 metres cubed per kilogram per second squared. This uncertainty is about one part in ten thousand. It might sound small to you still, but this is the best value that physicists have ever come up with, based on several very carefully done experiments.
Another problem with Newton's constant of gravitation is that there is no theory linking it with anything else in physics. Gravity sits apart from other forces. For example, we have known since Faraday's time the connection between electricity and magnetism, and, more recently, the connection between the electromagnetic forces and the weak nuclear force that acts in the nucleus of an atom. But, despite the efforts of people like Einstein, gravity still sits excluded. It really is a strange phenomenon.
For more information on the fundamental constants, see http://physics.nist.gov/cuu/Constants/index.html
If you want details of how things are measured, have a look at the rather technical paper of Mohr et al (2007) http://physics.nist.gov/cuu/Constants/codata.pdf