I have a Synology 1618+ and I’m very happy with it, not sure that I could describe it as budget though. There is a Plex app that comes for it and I have the majority of my dvd collection now converted and stored and they play very nicely on the TV downstairs. So worth checking that it’s easy to run what you want to use or there is a compatible alternative on top of the given management software that comes provided as warranty can be tied to using approved software.
One consideration I had was about trunking which is also supported so that if my PC is doing a lot of work talking to the Synology box we can still stream media due to the dual gigabit connections from the switch into the box and separate connections to the pc from the downstairs systems as well.
As pointed out above you should avoid raid 5, either raid 6 or another 2 disk redundancy, but I’d state that applies to any size sata disk because of the major difference in how SCSI and SATA disks behave with regards to data write verification. We discovered this when building large storage arrays during the period of time everyone was moving from SCSI to SAS (which are really SATA) that SATA doesn’t mandate a read after write in the disk controllers to ensure successful writes to unreadable sectors are relocated immediately to prevent subsequent data loss. This is required by SCSI, and some SATA raid controller cards also implement it to compensate, but you’d have to be sure it’s being done. Some filesystems may also implement this check, but the problem is ensuring the data isn’t just being read from the disk cache. Some disks lie. So even if you have some cheaper smaller SATA disks, raid5 is rather risky. And if you have a larger storage array it’s practically guaranteed you’ll hit one of these. With the 2 disk redundancy, a URE on a later read should result in a remapping with the missing data recovered from the other disks during a rebuild, the assumption being it’s very very unlikely to hit this error affecting the same block of storage across disks.