How precise can current technologies measure the mass of an object?

The most precise measurement of the mass of an electron was reported by Sturm et al in Nature 506, 467–470 (27 February 2014), quoting a relative precision of $3\times 10^{-11}$, meaning they determined the mass to better than $3\times 10^{-41}~\rm{kg}$.

If that is not the best, at least it gives you an upper bound...

Note that if you could weigh such a small mass directly with scales on earth, the force would be equivalent to the gravitational pull of a mosquito (mass 2.5 mg) on a grain of sand (0.7 mg) at a distance of about 6 million kilometers - about 17 times the distance to the moon...

Astonishing.

Acknowledgement: CuriousOne's comment got me thinking about the measurement of the mass of the electron, and led me to the above analysis.