BASIC
The best bang for your $, with minimum time investment is a Synology NAS. You slap some drives in there, plug it in and it just works. Click a few button on the web interface and you have it serving up across multiple types of local file sharing, with room to do other things like run a Plex server to serve up that media content. Combining this with a VPN, and the backups you're talking about can be done from anywhere. Without the VPN, machines can run LAN backups when configured from windows, and there is timemachine capability for the Macs.
ADVANCED
As for the Advanced setup, snag a server-class chassis 4U with the CPUs, and RAM inside from someplace like eBay, you'll likely spend between $500-$800. This is surprisingly cheap, and will probably cost you much less than the Synology. Pickup a compatible HBA PCI card with as many PCI lanes as your motherboard can put out in one of its PCI slots (like a LSI SAS3008) ~$50 , and the right SAS or SATA cables to attach to your drives, ~$40(?).
Next, Snag some inexpensive enterpise 3.5" drives from serverpartdeals.com for your storage setup ( $225 for a 20TB Seagate drive is really cheap, buy at least 2 , I recommend 5 so you can do ZFS RAIDZ2 ). Snag at least one SSD drive depending on what the motherboard supports or the operating system installation, NVME may be another option if you choose a modern motherboard.
Install Linux on the beast, and run ZFS. Use the /dev/disk/by-id/ names for the ZFS tank you setup.. Enable NFS for the Macs, install Samba for the Windows machines. Install Plex server for the media and you are READY.
If it's too loud, snag an enclosure the Trip-Lite ones are quite nice. I purchased a cheaper silencing cabinet and have regretted not spending more $.
References:
HBA Adapter
https://www.storagereview.com/...
note - this review is 10 years old ! The equipment is still good, but there might be slightly better HBA cards out there. Look for PCI Gen 4.
You may need to get the chassis separately ( $200 ):
Supermicro 2U Server 12 Caddy Bay 3.5 LFF E ATX Storage Chassis SAS2 6GBPS Rail
https://www.ebay.com/itm/15488...
HDDs:
https://serverpartdeals.com/co...
Silent Rack
https://tripplite.eaton.com/sm...
Good luck !