Would it be possible for a person to determine the volume of their body without using a water-displacement test? Since density varies with body fat and muscle mass, you couldn’t use volume = mass / avg. density.
What does Rabies have to do with it? 
Scared of water? Use tequila instead.
Seriously, your question is: what other methods are there for measuring the volume of a body of irregular shape and indeterminate density? That’s a toughie.
You could freeze the body solid and go over it with calipers for about three days. That would probably only get you a rough estimate.
You could chop it into tiny bits or liquify it and put it into a huge measuring cup.
Eureka! Liquid-displacement is by far the easiest method.
One could do it, carefully, with a gas, using the famous gas equation PV=nRT.
You’d have to seal a chamber of known volume V, introduce a fixed quantity n of gas, control its temperature T, and measure the pressure P.
Then introduce your subject and repeat the measurement. The difference in pressure should tell you the difference of the volume in the chamber, which would be the volume of the subject.
It’s not as easy as Archimedes’ method, because this time you’re using a compressible fluid (a gas) and the act of pumping gas into the chamber will affect the temperature, which will affect the reading.
I suppose if you could measure the temperature accurately, you could rely on math to simplify your apparatus, but the real problem is controlling/measuring the temperature of ALL of the gas. This is going to introduce some uncertainty in your measurement.
But on the plus side, you can even do this using offset pressures and volumes so that you don’t have to evacuate the chamber and expose your subject to a vacuum! 
And you thought you were speaking theoretically, bughunter:
This may be a copout, but filling a tub with a granulated solid, such as sugar, will get you a similar result, though maybe not as precise.
Or you could use pingpong balls and just count the number that fall out of the tub!