basically any PC with a recent (intel 10th gen and newer quad cores) CPU will work great for any normal media server build. You will just need enough space for your disks and some room to grow, the motherboard, cpu, ram and psu.
since you already have disks with media on them moving to a dedicated NAS OS will be a bit of a pain if you want some form of data protection. I'd definitely allow in the budget for at least 2 new large disks to start with. Personally I went down the unraid path as it allowed the most flexible disk mixing and matching, I could just throw whatever HDDs I had into it and all data was parity protected. it's not free but it makes for a good home NAS. moving existing data and re-using the disks is a pain as you need to start with enough space to dump a whole disk to, then wipe that disk then add it to the array, then repeat for all of your disks, this can take days but it works and gets your data loaded and parity protected with a minimum number of new disks required.
Freenas, now called Truenas is an excellent option but it will be less flexible in adding disks that arent the same capacity. you cant just buy one HDD and drop it in to expand in the future, you tend to need to plan it out a bit more, but it is extremely fast and very reliable. so it's free but can cost more in the long run.
If you like to tinker you can just run something like ubuntu and set it all up from scratch, or there is one called Xpenology, which is a clone of the synology software, it is very easy to use and reasonably flexible.
You can just plug the HDDs into the motherboard if it has enough ports, but I'd recommend getting onto eBay and getting yourself a SAS HBA card and sas-sata breakouts, there are sellers that have them as combo kits just for this purpose.
My first couple of server builds used the motherboard ports and the SATA controllers died pretty quickly, then I got a LSI 9211-8i, than added a sas expander for more ports, and more recently a newer 9300-16i card that will do me forever.