What's the precision of a modern Inertial Measurement Unit, and how is it obtained?
Let's have a look at documentation of actual modern IRS. It says (on page 6) that if GPS data is lost, it will maintain RNP0.1 for 8 minutes, RNP0.3 for 20 minutes and RNP1 for 2 hours. That means that in 8 minutes the error is no more than 0.1 nautical mile with 95% confidence, in 20 minutes it is no more than 0.3 nm with 95% confidence and in 2 hours the error is no more than 1 nm again with 95% confidence.

For general description how it works see What is an Inertial Navigation System? How does it work? and of course the appropriate Wikipedia article. Basically it measures acceleration and rotation (with gyroscopes) and integrates it over time. Modern systems do the calculations digitally with sophisticated filters to smooth over some errors.