What I have (and what I would suggest for you) is a dedicated "storage machine" which is a PC case, motherboard and power supply and hard drives in a RAID configuration.
No need for a video card since the motherboard will probably have some sort of video output and you can also remotely administer the storage machine via SSH.
My box has five 4 TiB drives in a RAID6 configuration (giving me 12 TiB storage space that can survive two failed drives). The box runs Linux server and it's super easy to setup.
As far as Windows, the storage drive looks like any other "drive" in the system. To backup stuff, simply copy it to the storage machine.
Here's what it looks like in a Linux file manager:
(click pic for larger)
It's similar in Windows. Just drag-n-drop stuff from your computer to the storage machine. And, being on your network, anyone in the family can use it, and you can even access it off-site.
If you're interested, I can help you with instructions how to set it up. It's super easy.
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