If the accuracy of measurement of the earth's volume is not known, how can scientists assert that the earth is not expanding?
In short, the current volume of the earth is known to great precision and modern measurements show no significant expansion at meaningful scales.

Facing 20th century evidence that points on the globe -- such as the edges of continents -- that are now distant from each other were once adjacent, some scientists argued that the only explanation was the ongoing expansion of the earth's surface. That argument collapsed by the 1960s, as overwhelming evidence accumulated for the theory of plate tectonics -- that is, roughly, there are relatively rigid continental plates that move across the surface of the earth, with the ocean floor between them created at the mid-ocean ridges and destroyed by subduction under the continental margins. Broad calculations showed that the volumes of the two processes were close enough to cancel each other out.

In the modern era dependent on the accuracy of satellite positions for GPS, sea-level rise measurement, and other detailed geodetic measurements, there was a need to resolve the question of whether the earth's volume was changing in a more conclusive and precise way.

One of the studies that addresses that is the 2011 paper Accuracy of the International Terrestrial Reference Frame origin and Earth expansion, published in Geophysical Research Letters, a journal of the international American Geophysical Union. Most of the paper discusses the technical processes and calculations, but it concludes:


  A possible secular change in the mean radius of the solid Earth has
  long been suspected from various scientific disciplines. The problem
  was not settled due to lack of a proven theory and the inherent
  difficulty in actually determining such a change. By using the data
  combination and the simultaneous global inverse approach, the issue of
  aliasing from the geophysical processes due to network sparseness is
  largely overcome here. In fact, our uncertainty in the mean expansion
  rate determination is dominated by an ITRF [International Terrestrial Reference Frame ] scale drift uncertainty of
  0.16 mm yr−1 estimated from comparing VLBI [Very Long Baseline Interferometry] and SLR [satellite laser ranging]. In conclusion, no statistically significant present expansion rate is detected by our
  study within the current measurement uncertainty of 0.2 mm yr−1.