I have a cylindrical container with a radius of 60mm and a height of 58.8mm

when I work out the volume I get ~ $665000mm^3$ which is something like $665 m^3$.

This container is really small. I can't understand how 665m can be a measure describing something this small. Can someone help me better understand what I'm looking at here?

Your conversion factor is wrong. A cubic meter is NOT 1000 cubic millimeters. You have to start with "1 meter=1000 millimeters", then cube each side, so you get 1 cubic meter=$1000^3$ millimeters, or 1 cubic =1000000000 cubic millimeters.
You are not accounting for the fact that there are $(10^3)^3$ $mm^3$ per $m^3$.