I found a 12 V UPS, a Mean Well AD-55A. It outputs 4 A at 13.8 V DC and a small battery can be attached (charge 13.4 V, 0 - 0.23 A). Output voltage is adjustable between 12 - 14.5V.
It has of course to be built into a box to be electrically safe. I had an old monitor switch box that was suitable. A few holes were cut to add a fuse and five "DC plug" 12 V outputs. Add a small combined voltage and current meter and we have this:
You have to find the correct DC plugs that fit into the equipment. Be sure to check the polarity!
Installed in the cabinet together with the small lead battery from the previous UPS:
In the top is the network patch panel. Middle right is the big cable modem (DOCSIS). Middle left is the router, an Ubiquiti Erlite-3. A fantastic router. I mistakenly run it on 5 V when I bought it. It didn't mind, but for sure 12 V is better. At the left on the inside of the cabinet door is a small 8 port Gbit network switch. In the bottom is the battery and UPS. Note that there is no Wifi access point in the cabinet. It wouldn't work well inside the metal cabinet.
I mostly did this just because I liked to do it. There is a bit more space in the cabinet for air flow and it is definitely not as hot as before. Some electricity is probably also saved, although I didn't do any measurements. The cost was mostly the power supply/UPS module. It was 28 € + VAT + shipping.
It took a lot of time to make the DC plug patch cables, though. But I don't know if you can get them ready built from somewhere.
If I had a large room for the equipment, I would definitely use a large lead battery with a regular car battery charger. Then use a 12 V - 12 V DC power supply to drive the equipment.