For what it is worth, I’d highly recommend using some form of software defined storage pool solution rather than a hardware RAID card. it tends to be more reliable.
If you are comfortable with Linux/Unix command line you can roll your own using Linux or FreeBSD and OpenZFS. If you are not quite as comfortable with this TrueNAS (the old FreeNAS) has an appliance OS release with a web management interface which is pretty nice and usable.
The “most right” way to do it from a reliability perspective is to use proper server boards and enterprise SAS HBA’s in a server case that has a backplane that supports hotswapping, but as a starter you don’t really need to go that nuts. (You can gradually migrate there over time like I have)
From a platform perspective - however - I recommend making sure whatever you get supports ECC RAM.
A couple of years back I upgraded my router build using a Supermicro X12STL-F board and a 11th Gen Rocket Lake based Xeon E3 CPU. It wound up being a pretty cost effective solution for me, and I remember thinking that if I put a SAS HBA in it, it could be a nice lightweight ZFS box.
Don’t be afraid of used server pulls from good recyclers on eBay. It’s a great way to get a fantastic deal on enterprise hardware, and I have never had an issue. The hardware tends to be well taken care of (compared to used consumer stuff which is often mistreated by idiots)
But there are many many options, and configuring and building them is part of the fun!
Just a slight caution. There are many “barebones used servers” out there that are terrific deals for what you get. Just keep in mind that these are business machines that usually are at home in server rooms. The drive noise is going to be the least of your concern in those things. The fans alone can often wake the dead.
I got a barebones server like this years ago based around a Supermicro SC846 case, and it has been great, but it took some modding to make it “home compatible” from a noise perspective, and even so I keep it in a rack far away from my office.
I once got a HP DL180 G6 back in 2014 and attempted to do the same, and I utterly failed. That thing sounded like a jumbo-jet taxiing on a runway. With it in my basement at the time, with two doors closed in between, the noise was still loud enough to be bothersome in my bedroom two stories up 
Seriously, not kidding about the jumbo jet thing. Watch this video to see what I mean:
(Not my video or my exact server)
In my case, because I populated one of the PCIe slots with a PCIe card (SAS HBA) that the servers BMC/IPMI didn’t recognize it went into 100% fan mode permanently to make sure it was cool enough, and it had 8x little 80mm fans that went up to 18krpm I think. It was totally nuts.
So, you are probably going to want to build something yourself when it comes to managing noise.
My backup Workstation in my office has 6 HGST 7200rpm SAS drives in it, and it is pretty good from a noise perspective. I built it into an OG Phanteks Enthoo Pro case. But there are many options out there.
For what it is worth, here it what it sounds like with 6x 4TB HGST drives in it. Most of the noise is fan noise.
Though, this one is only sporadically powered on, so I didn’t get obsessive about noise. If you have a noise focused build approach you can probably do better.
Also, of course I realize that with no baseline, it is impossible to tell exactly how loud that is. For a qualitative description, if I am sitting right next to it with the cover put back on, it is audible, but not really bothersome. From across the room ~10ft away it is only borderline audible.
In general, apart from occasional head seeks, I have found that modern hard drive rotational vibration is pretty low key. Usually its the fans needed to keep the hard drives at reasonable operating temperatures that create most of the noise.
Good luck. I’m curious to see what you come up with 