What's the precision of a modern Inertial Measurement Unit, and how is it obtained? This recent comment reports that:


  the IMU on new (plane) would localize the aircraft to within 3 feet
  after a cross-country flight, without any GPS input other than the
  starting location.


I somewhat doubt about this statement, at least for an IMU based solely on inertial measurement: over the duration of a flight, I fear much more error accumulates.

So what's the precision of a modern Inertial Measurement Unit over say the duration of a flight, and from what sources is that obtained? If some source (in particular, GPS) becomes unavailable, how does it degrade that accuracy?
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.