To what accuracy and precision do we know the volume of the Earth? Satellites such as GOCE and GRACE measure the geoid with unprecedented accuracy.  Altimeters can determine local surface elevation with millimetre-precision.  This makes me wonder: to what precision and accuracy can we measure (or calculate) the volume of the Earth?

I'm a lay when it comes to solid Earth, and I find it surprisingly hard to find even the order of magnitude of an answer ($10^0$ metre?  $10^3$ metre?  $10^{-3}$ metre?).  Wikipedia is extremely limited on the question and links to a surprisingly poor NASA factsheet that doesn't include any error estimates, cites no sources, and is imprecise in providing context or defining what it's talking about.  

One can calculate the volume of Earth using semi-major and minor axes obtained from reference ellipsoids, but I have no idea how large an error is introduced by this approximation.

What are the precision and accuracy do we know the shape of the Earth?
One way to approach this is to treat the Earth as an oblate ellipsoid. This would mean the errors arise from the uncertainties in the Earth's equatorial radius and the flattening. From Groten, "Fundamental Parameters and Current (2004) Best Estimates of the Parameters of Common Relevance to Astronomy, Geodesy, and Geodynamics," Journal of Geodesy 77:10-11, 724-797 (2004), the Earth's equatorial radius in the mean tide system is 6378136.72±0.10 meters, and the inverse flattening is 1/f = 298.25231±0.00001. The uncertainty in volume due to the uncertainty in equatorial radius is about $3 V \frac{\Delta a}{a}$, where $V=\frac 4 3 \pi (1-f) a^3$, or about 51,000 cubic kilometers. The uncertainty in volume due to the uncertainty in inverse flattening is tiny in comparison, about $V \frac {\Delta 1/f} {1/f (1/f - 1)}$, or about 120 cubic kilometers.

That 51,000 cubic kilometer figure sounds like a lot, but it's not. It means the volume of the Earth can be expressed to seven places of precision.