You might be able to use something like this, PoE splitter. Other brands might offer you a smaller size, such as Smaller PoE splitter.
Or, you can set up a solar charge setup with batteries ofcourse.
Or, you can even put in a Li-ion battery - Measure the current draw and you can get the run-time based on the battery specifications. If that is up to your mark.
I don't mind tearing apart an ethernet cable myself for the two DC PoE leads. My knowledge is a bit limited here, but I'm assuming I'll also need some sort of voltage regulator to avoid leading 24V into the Arduino.
Can I use something like this?
And is it feasible to run PoE over 10 meters?
Solar charge is not an option as it is indoors. I can run 230V AC there if needed, but I assume this will only make it more complicated. The main goal is to keep the size down.
I have a security camera system that uses PoE for the cameras. The ethernet cables that come with it are 100' long so your 10 meter length will be no problem at all.