They (DSLAM/MSAN?) usually have battery backup, I lose grid power all the time but my internet still works without interruption.
https://www.facebook.com/NS.net/photos/a.699290313417141/944921128854057/?type=3
"Typical outdoor MSAN cabinet consists of NarrowBand (POTS), BroadBand (xDSL) services, batteries with rectifiers, optical transmission unit and copper distribution frame."