The battery idea sounds like (almost) a perfect solution. Small devices, like an ethernet switch eg, could be run off a 12v motocycle battery, which would last for weeks. One issue may be that a 12v car/motorcycle battery is 13.8v when fully charged, so I'm not sure of the tolerance of the input circuitry, although a 12v wall wart outputs more than 12v anyway when not under load, although it will drop when the device is powered up. The battery will not have that voltage drop. Batteries are also, relatively, cheap.
I will buy a battery to power my switch/NAS/Streamer and post the results here in a few weeks.