Really, you should be looking at 3 different devices for the different tasks of: DVR, NAS, Plex. Your budget looks doable, although I have to admit at first glance I thought no way!
Run plex on a dedicated HP290(they have been going on sale for ~$100)
If you have any old computers laying around, chances are you can get a nas running unraid set-up for file storage, Or check out some of the sample NK4 Builds as low as ~$175
Blue iris is the way to go for cams, but this might take up your whole budget depending on how many cameras and megapixels you want to add.If you’re only running 4 coax cameras, another HP290 might be able to handle that, but idk how much you would be able to add to it, you could get a faster, but older processor to give yourself some room to grow for ~$200 like this.
tldr:
NK4 for NAS ~$175 (Driveless)
HP290 for Plex ~$100
HP290 for BlueIris (Confirm usage/megapixels!) ~$100
or
Older I7 w/ quicksync for room to grow with blue iris ~$200-300
Note I didn’t include any drives for the NK4, but if you hang around in the discord on here you should be able to pick up some cheap dense disks pretty quickly.