Copernicus deduced the scale of the Solar System. He did not figure out how big the Solar System was in kilometers, but he did figure out the relative size based on the size of the orbit of the Earth (i.e., on the Astronomical Unit).

Relative Scale of the Solar System

Armed with nothing more than the sidereal periods of the planets, Copernicus deduced the relative spacings of all of the planets (the observable ones that is) of the Solar System. In fact, for the planets whose orbits were smaller than the Earth's, he did not even need this information.

Inner Planets

Outer Planets

So, given some rather fundamental numbers about the planets, Copernicus based on his helio-centric theory was able to deduce the size of the Solar System.

RESULTS

Planet  Copernican Estimate (A.U.) Current Value (A.U.)

Mercury      0.38                       0.387
Venus        0.72                       0.723
Earth       1.000                       1.000
Mars         1.52                       1.52
Jupiter      5.22                       5.20
Saturn       9.17                       9.54

The results are amazingly good.