Answer
Yes. It is possible.
Work Step by Step
$F=G\frac{mM_{e}}{R^{2}}$
m = mass of an object on earth's surface.
R=radius of earth at that point.
Again, $F=mg$
Thus, $mg=G\frac{mM_{e}}{R^{2}}$
or, $g=G\frac{M_{e}}{R^{2}}$
So, mass of the earth $=M_{e}=\frac{gR^{2}}{G}$
Since all the quantities on the right side are known, we can find mass of the earth.
