The UPS contains a battery. At 6 or 7 years, you may have reached an "end of life" condition for the battery itself. (Five years is considered a normal time-to-failure for auto batteries, even if you pay for the extended warranty.) The beeps may mean that the battery is not holding stable voltage and/or is overloading the charging circuitry in the UPS. (Just a guess, of course.)
Another possibility is that the power circuit (input) is seeing transients that trigger the warning, but don't last long enough to initiate a shutdown cycle.
A third possibility would be a problem with the charging/sensing circuitry in the UPS.
Product documentation that came with the UPS should have included some explanation of what various beeps and flashing lights mean(?).
The UPS manufacturer's website may have a downloadable Operator Manual and/or Service Manual? Or an on-site FAQ?
If the problem is with the battery, it probably can be replaced, but you really should check the manufacturer's recommendations on how to get to it. You likely will need a "gel-cell" (non-spillable) battery, which will be similar in price to, or possibly a little higher than, a "spill-resistant" motorcycle battery. An exact replacement would be preferred, but you might find a "close-enough" from one of the "handicapped mobility vehicle" shops.
If the problem is short duration transients due to the pending failure of your refrigerator, you're on your own about repairs.
If it's due to transients caused by the squirrels that ate your pole transformer, it should at least make a good story.