One approach is scale-dependent (divide the area into a grid of square kilometers, sort by population, pick the square kilometer where the median person lives.) but you would get different answers if you picked different scales.
The median of "How far away is your nearest neighbor?" seems parameter free at first but it is absurd if you live in an apartment. (e.g. ignores all the space around your apartment.) I guess you could look for the median (or other decile) of "What is radius in which N people live?" where N might be 100-10,000 but then you're back to having a spectrum instead of a single number again.
Maybe you can't avoid that.