How is the most accurate value of $G$ measured? How do modern scientists update the measurement of $G$, the gravitational constant? Is CODATA the authority on this measurement and the experiment?
 A: There is a recent paper in which $G$ has been measured. 

We obtain the value $G = 6.67191(99) × 10^{−11} m^3 kg^{−1} s^{−2}$
  with a relative uncertainty of 150 parts per million (the combined
  standard uncertainty is given in parentheses).

Most measurements of $G$ are based on a torsion pendulum as in the original Cavendish experiment. The paper doesn't quite agree with those measurements and it is not clear yet which is correct.
In this experiment the wave nature of atoms is exploited to do a very precise measurement. As in a Michelson interferometer. On of the paths acquires a phase that is compared with the reference phase. In this case, the gravitational interaction is what causes the phase shift on the atoms.
A: The last measurement I know is one made in 2007 using an atom interferometer  http://www.sciencemag.org/content/315/5808/74.abstract. They report 
"a value of G = 6.693 × 10−11 cubic meters per kilogram second squared, with a standard error of the mean of ±0.027 × 10−11 and a systematic error of ±0.021 × 10−11 cubic meters per kilogram second squared." 
Also, this year under the assumption that the physics of type Ia supernovae are universal, analysis of observations of 580 type Ia supernovae has shown that the gravitational constant has varied by less than one part in ten billion per year over the last nine billion years
A: The latest (2018) measurement is
$\boxed {G = 6.67430(15) × 10^{−11} m^3 kg^{−1} s^{−2}}$ with an uncertainty of 22 parts per million.
See the paper, "Measurements of the gravitational constant using two independent methods".
However, there is concern regarding conflicting measurements.  Efforts are underway to reevaluate the conflicting results of measurements.
See also Wikipedia.
And related: How precise are the observational measurements for the speed of gravity?
A: It’s generally accepted that the value for G, as at autumn 2022, is 6.6742 +/- 0.0003 x 10^-11 m^3 kg^-1 s^-2.
Gravitational waves cause problems, quite aside from any quantum entangled occurrences. All models are wrong, but some are helpful.
If you think about these accuracy limits for just a few seconds you’ll see that we can’t be accurate in numerical calculations for astronomical purposes to more than 5 significant figures. In other words we have an inaccuracy of one in a million.
Not so amazing then that we really can hurl a satellite the size of an SUV and hit an asteroid the size of a football stadium at a distance of almost 7million miles (Pythagoras and sines/cosines will help you here!)
