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Depending upon the maker and model of the black battery backup box, you may need this or that specifically.
But it sounds to me, all told, as if you have to have it serviced, or at the very least replace the battery in it. All rechargeable batteries in electronic devices nowadays have a limited lifetime and do fail eventually, in a year and a half to several years of active use or inactive disuse, or sometimes sooner.
That the battery backup box ceased (evidently) to provide power supports this assumption, that the battery in it most likely went dead, and that was what the beeping (a self-detection failure alarm) was about.
The battery backup box, at a guess, is a 'UPS' or Uninterruptible Power Supply. Those stabilize power supplied to devices by passing power from the wall into a charging circuit for a battery, then taking power for the device running off it, from a charged battery. When the battery gives out or dies, then the device won't run because the battery won't supply power enough to it.