I have an APC UPS in my home system, which I modded and externally connected 2 full-size Optima deep cycle gel batteries to it externally, and I installed UPS-NUT tools via Funplug onto the DNS-323, and it is the NUT master which also controls 2 slaves (other servers) in which if the power goes out it can withstand an 8-10 hour power loss, but with the UPS-NUT tools, if the battery voltage goes below 10.5 volts the master (DNS-323) will safely shut down the other 2 slave servers, and then shut himself down gracefully.
My system is unique in that my 2 servers actually run using 12v DC power supplies, and all my wireless APs, switches, etc, are also running on the low voltage power bus via a DC-DC stabilizer (mainly only use the APC UPS as the sensor to run UPS-NUT tools), but for the moment I do run the DNS-323 off of the AC power adapter so it does go through the inverter circuit of the APC UPS. I also disabled the internal charger inside the APC UPS and charge the batteries using a standalone power supply/ battery maintainer which can rapid charge up to 60 amps for fast recovery.
But in your case, you could get a simple APC UPS same as mine (or use the one you already have), and just wire it to a larger external gel battery (gel batteries don't emit flammable gases and are safe for indoor use), and just by doing that alone you would most likely be able to run easily all night on that charge, and one could configure the DNS-323 for safe shutdown using UPS-NUT tools if desired to go to that extent.
If that sounds too complicated then you can just do what the other guy suggested and pull the front cover off so you can see the actual little micro-button, and push that directly to check and see if it is faulty. If it is, you can go to Digi-Key website and order a replacement one, and it could easily be soldiered into place. I could fix it for you if you needed someone to do it, and could handle the downtime for a few days turnaround, or you could take it to a local electronics repair shop and they could do the soldiering as well.
The last option is to just switch to cloud storage like Google Drive (if you have a fast enough internet connection), as well..