Do I need a dedicated machine to do this?
Depends. Some games are more demanding to emulate than other. Like most emulation, RPCS3 is CPU-intensive, requiring a decent CPU, particularly one with good single-threaded performance. I would recommend a 6-core or 8-core, preferably Intel Skylake (aka 6000 series) or AMD Zen 2 (aka 3000 series). You can get away with a quad core or something older, but performance will be spottier in some games.
GPU is less important. It only really needs to be capable of rendering graphics from mid-00's hardware, so any dedicated GPU that supports OpenGL 4.3 from the last 10 years is plenty, tho you could probably get away with one of the newer intergrated GPUs. But a GPU that supports Vulkan (i.e. RX 400, GTX 900 or newer) is recommended, because performance will generally be much better. You'll also want something beefier if you plan on rendering the games at resolutions higher than 720p, i.e. 1080p, 1440p, 4K and up. I seem to remember you have a RX 480, which is more than enough.
All in all, any decent gaming pc from the past 5-6 years will do just fine.
Do I need to download digital games or run my real PS3 discs in a DVD player?
Putting a PS3 disc in a DVD player wouldn't work to begin with, cuz PS3 uses Blurays. Tho RCPS3 can't run PS3 games directly anyway, whether from a Bluray player or a ripped ISO, so that's a moot point.
What you need to do is install some custom firmware to your PS3 that'll allow you to install the whole game from the PS3 disc to the PS3 hdd, then use whatever external usb drive you got lying around to transport the game files from your PS3 to your pc. There are plenty good resources for how to do it, starting with the RPCS3 website. Ripping your own games is kind of a time-consuming process tho. The other option is to of course visit a certain bay, tho for obvious reasons I can't help you with that.
Aside from that, the only other thing you'll need is the PS3 firmware, which you can download for free from Sony. Iirc, there's a link to the right place in the Quickstart guide on RPCS3s website.
Does it see my hard drive for storage?
Yes, RCPS3 will run the games directly from whatever drive on your pc, whether mechanical hdd, ssd or even an external flash drive.
Still, gives me hope for my PS3 games. I still have 2 old systems. Oldest from 2009. It's going on 13 years in age. This gives me hope that when they break, I may still use my games, especially as Sony is argued to not be supporting the PS3 anymore. (No BC for PS3 on PS4 or 5).
Keep in mind RPCS3 is very much a work in progress. A lot of multiplat titles will work and can be played start to finish, with a little bit of tinkering in the emulator settings and a workaround here and there. But the exclusives are a lot more hit and miss. Some will run, but most don't yet. Or require a custom fork of the emulator to work, and even then tend to crash a lot and exhibit lots of glitches. Or they technically run just fine, but can't be played properly because they use a PS3 feature the emulator doesn't support yet e.g. Sixaxis.