Everyone wants a server running at home, don’t we?
But not everyone has the geek powers to rebuild an old machine, configure it, secure it, poke proper holes in the fire wall and finally maintain it.
Not every server has to be run 24/7 and not every server has to withstand big loads or even be public to everyone.
So start small. A mobile phone on 3G is outside your fire wall. It might not be on your LAN even. It is low powered but has quite hefty innards to crunch numbers and is inherently good at networking.
Use some kind of dynamic DNS and start the app manually or automatic when connected to the charger.