A fire heated distiller like Ville Niemi's answer is a good idea, but if the fuel and constantly tending the fire aren't practical, you could try a solar distiller. Shouldn't need any space-age materials, just some glass & metal (avoid lead solder!) or wood containers to hold water in.

[![Solar still image][1]][1][The Solar Water Still Challenge](http://www.techedmagazine.com/node/2609)

The guy in the videos on [this page][2] got 3.13 cups of water per day from 3.3 sq.ft of "glass area," or about a cup of water per day per sq.ft of glass. 

Or a more basic "hole in the ground" still like this may work too, you may not have access to clear plastic sheets, but a clean dark cloth or tarp *may* work also (and you don't have to worry about space-age plastic chemicals leeching into the water either).

If there's enough space in people's homes in the city (no high-rises then) they may each be able to have their own solar still & make their own water, no fuel or fires required.

[![enter image description here][3]][3]


  [1]: https://i.sstatic.net/ORaU4.gif
  [2]: http://diy-alternative-energy.com/build-a-solar-water-distiller/
  [3]: https://i.sstatic.net/dPoeq.gif