Retro ROMs are usually small. Videos can get quite large though, on the order of ~100GB per movie if you are storing 4K Blurays.
I personally bought a couple > 20TB HDDs off of serverpartdeals.com and installed them in my gaming PC so now it also functions as a small NAS. Because it’s only on when I’m using the PC, the electric bill is not too bad. But it’s worth doing the math to see what your average kW/hour usage is. Wattage monitors are pretty cheap.
If you specifically want a lower-power NAS in a separate machine, this will require a bit more research, and they can get pricey. I highly recommend using ZFS though.
If you’re OK using a cheap, low-power mini PC as a home server and/or gateway, I can recommend the BeeLink EQ12. Mine is currently running 24/7 attached to a Hasivo 2.5Gb switch with PoE powering my WiFi AP.
There are also options for connecting large external HDDs to a mini PC, but you would be compromising throughout via some SATA adapter.
This person is no superstar.