Why is the resolution or measurement uncertainty of $G$ so bad? I sorta know how the Cavendish apparatus works, and I know that "Gravity is an extremely weak force" for subatomic particles in comparison with the E&M or nuclear forces that act on these particles.  So I understand why such a measurement is extremely difficult and, in a single experiment why such a measurement would have a relatively high relative standard uncertainty, $4.5 \times 10^{-5}$, (or relatively few significant digits, ca. 5).
But if a quantity being measured is constant over time, I would normally understand that repeating an experiment many many times over a long period of time, and averaging the values, that should decrease the relative standard uncertainty with any measurement.
If I were measuring a DC voltage that had some noise added to it, with an analog-to-digital converter of relatively few bits resolution, I can add some known dither noise to the DC voltage and run the measurement for weeks, get millions or billions of measurements of that DC voltage, and average these measurements to get a value of increased resolution greatly exceeding the resolution of the A/D converter.
Why can't this U Washington experiment be set up and run continuously for weeks or months or years, to get a zillion slightly different values for $G$, and average those zillion values to get a good value of $G$ with 8 or 10 significant digits?

 A: 
I would normally understand that repeating an experiment many many times over a long period of time, and averaging the values, that should decrease the relative standard uncertainty with any measurement.

First, your understanding is, well, wrong.
This happy situation, where averaging decreases the type A uncertainty component, does happen only when the random measurement fluctuations can be modelled as white noise, that is, when they can be modelled as uncorrelated random variables.
Unfortunately, most of the noise processes that occur in experimental apparatuses during long-term measurements cannot be modelled as white noise. White noise is dominant for short-term measurements but, in the long run, noises with long-range correlations appear (e.g. flicker noise, random walk and, more generally, power-law processes).
For noise with long-range dependence (or long memory) it is no longer true that the statistical uncertainty decreases with time. For instance, it can remain constant, as with flicker noise or, actually, it can even increase, as with random walk. A good starting point to understand this problem is [1]. 
Typical noise sources of the above kind are the electronic noise of the measurement equipment, the seismic noise coming from the environment and the mechanical noise of the experimental apparatus (e.g. this is a well-known problem in gravitational wave detectors).
The consequence is that in any real experiment there is somehow an optimum measuring time (which, sometimes, can be quite short) at which the type A uncertainty reaches a minimum value and above which it no longer decreases.
In addition to this, in such kind of experiments, the ultimate measurement uncertainty does not depend on the type A uncertainty component but on the type B uncertainty component, that is, the uncertainty component which depends on the non-idealities of the experimental apparatus that lead to systematic errors. Examples are the inaccuracies in the determination of geometrical dimensions, the inhomogeneity of the test masses, etc.
For what concerns your idea, expressed in a comment, of averaging across different experiments, this is actually done. The recommended values of the fundamental constants given by CODATA that you can find in tables are actually averages across the state-of-the-art experiments. It should be noted, however, that some experiments can be affected by systematic errors with the same direction, and averaging would not reduce the effect. This probably actually happened in some of the past $G$ experiments (before the 1990s), because it was later found that the inelasticity of the fibre used in dynamic (time-of-swing) experiments could produce a systematic error [2].
[1] J. Beran, Statistics for Long-Memory Processes, Chapman and Hall, 1994.
[2] K. Kuroda, "Does the time-of-swing method give a correct value of the Newtonian gravitational constant?", Phys. Rev. Lett., 75, 2796, 1995.
A: The apparatus may (does) have systematic biases (errors). You can't get rid of those by doing the same experiment over and over again.
In analyzing their experiment, they will have determined what the maximum value of these systematic errors is. For example, because of the tolerances on machining the parts of the apparatus, or their limited ability to control the temperature of the equipment.
Presumably the designers of the experiment have already chosen to run the experiment enough times that the residual random error is much lower than the limits they have achieved on their systematic errors.
