Are you planning to use this only for media storage, or other tasks as well?

Many dedicated NAS boxes from the likes of Synology and QNAP come with low-powered CPUs that in addition to handling media storage can do many other useful things. For example - running various self-hosted apps like password managers, photo organizers, even office suites. Those do have many downsides as well. They use proprietary hardware and software, so you are at the mercy of the manufacturer if they suddenly decide that your NAS is too old for them to support. The CPUs in them are not as powerful as consumer-grade ones found in most desktops, and they don't usually come with a lot of RAM. They are also quite expensive for what they offer - for example, a very basic 4-bay enclosure with 2GB RAM and Realtek 4-core CPU costs $370, and that's without the cost of the disks. For that kind of money you can build a desktop from used server parts that will be ten times as powerful, and more versatile since it will be able to run any OS that you want. Granted, it will not be as small or power efficient.

If you just need a turn-key off-the-shelf solution, a prebuilt NAS may be the way to go. If you want to tinker with hardware and experiment with software however, self-built system with used parts may be better suited for you. Most homelabbers don't start off with a rack full of equipment, they grow their hardware organically. You have outgrown your external enclosure, so now you have several choices. For prebuilt NAS appliances Synology and QNAP are not the only choices; there are many other lesser known in this category - Asustor, Terramaster, Buffalo, to name a few. For building a server on your own I would recommend doing some research before going on a shopping spree - https://serverbuilds.net is an excellent resource.

That said, I'd echo others' answers. RAID is not backup. It offers some resiliency from disk failure, but that shouldn't replace a proper backup strategy. Synology offers various RAID levels, including their own take on it that allows for different sized drives. For self-built option there's UnRaid which handles resiliency differently, but requires an entire separate drive for parity.

In terms of hard drive choice, it all depends on how comfortable you are with possible data loss. If you back up your critical data (e.g. family photos, important documents) and don't care about downloaded media (you can always re-download it) then go ahead and use whatever - shucked, used, etc. If you absolutely cannot handle any data loss, then new drives made specifically for NAS appliances are the way to go. Enterprise level HDDs are an option also, but those mostly use SAS interface instead of SATA, and require specialized hardware (HBA adapters).