Weaver has three lines bonded, so three modems and one FireBrick router would need to be battery powered. As for trickle-charging, sheddyian might be able to advise about solar-powered charging of lead-acid batteries.
I had been meaning to chip in to this thread, but time was against me
Previously (can someone find the thread?) I'd tried running my router off of a 12 volt drill battery, and compared the ADSL2+ results to running from mains. I wasn't suffering from any known interference problems at the time. I found there was no noticeable difference, even when isolating the router from ethernet and relying purely on wireless to check the stats.
Similarly, when I
was suffering from REIN on my ADSL2+ connection, where the synch would drop and resynch much slower for hours at a time (and bit/tone allocation severely affected), running from an electrically isolated battery would not improve matters for me.
[This REIN interference is now, thankfully, gone, I still do not know where it came from though assumed it was an indirect neighbour with a noisy LCD second TV that was switched on a few times each week]
I am currently running a Raspberry Pi web server from two standard car batteries wired in parallel to give 12 volts but increased current/ length of service.
The batteries are charged from a 20 watt solar panel that I bought on ebay for about £30. If you're going this route, over-estimate the input current to charge the battery. It only needs a few overcast days for your supply to be on a knife edge. I've yet to run my setup through a full winter without giving a sneaky top-up by connecting a mains charger to the batteries because I was anxious.
That said, running a router purely on battery power, and isolating it from ethernet cables (which could carry mains interference from switches or other devices) is probably the best you'll get for connection quality. And in my experience, it made no difference from running everything off the mains.
Ian
