The most perfect manmade spheres constructed to date are the fused solid quartz gyroscopic rotors built for NASA's Gravity Probe B spacecraft. There are four spheres onboard, each measuring 3.81 cm (1.5 in) across. Their average departure from mathematically perfect sphericity is 1.8 X10-7 of the diameter. This means that, if scaled up to the size of the Earth, the maximum height/depth of topographic features would be 1.5 m (4.92 ft).
Gravity Probe B was launched on 20 April 2004 and will spend 16 months testing Albert Einstein's General Theory of Relativity.
Comments below may relate to previous holders of this record.