I really hate batteries, while they are reliable way of getting things off the grid, I find well designed low noise power supplies based on AC to perform better, and to be way less of a PITA. On my router I built a custom ultra low noise/impedance supply, starting with a very high performance Daitron low leakage SMPS (yes, yes, I know, everyone will now suggest all SMPS are evil, just not the case though). Unfortunately the really good Daitron supplies appear to be unavailable now. Anyway, i dialed the output of the Daitron up ~14 VDC, and then followed it with a moderate size electrolytic cap, and a Belleson SPZ discrete regulator. I also made sure to ground the DC output to AC ground to shunt the high impedance leakage current. The result: µV levels of noise, and ultra low output impedance to my router. I compared to a full linear also with Belleson regulation and there was no difference.
BTW, I would not recommend those battery packs mentioned earlier in this thread (the ones marketed for charging up phones and powering other portable devices) as their outputs are realized via noisy switching DC/DC converters.