For pretty precise measurement you use linear acceleration of the body with fixed force (say, spring pulled until its force reaches nominal value) and then you measure its speed when launched.
Kinetic energy $ E={{1}\over{2}}mv^2 $ will be equal to potential energy of the "launcher" (which can be easily calibrated by launching an object of known mass and measuring its speed using the same assembly).
Now, given known energy and measurable speed we can calculate the mass:
$$ m={{2E}\over{v^2}} $$
For example, assume the launch assembly is $ 5 kg $. You launch $5 kg$ weight out of the assembly and measure the distance it travels over the course of 1 second. It goes $ 10 m/s $;
$$ E= {{1}\over{2}}(5kg+5kg)(10m/s)^2 = 500J $$
that's the force of the assembly.
We launch a human, and the result of measurement is $ 4 m/s $.
$$ m = {{2 \cdot 500J}\over{(4m/s)^2}} = 62.5kg $$
Substract the known $5 kg$ of assembly, and the human mass is $57.5 kg$.