If your router has a USB port, it probably has an option to share a HDD over the network. Just put the HDD into a USB enclosure and plug it in. Then configure the router for shared storage.
It won't be very fast - older routers typically only achieved 10-15 MB/s with a HDD hooked up this way. Newer ones usually manage 30-40 MB/s. But that should be fast for basic file sharing and backups. The first backup will take forever, but if you make sure subsequent backups are incremental or differential it won't take anywhere near as much time.
After that, what you want is called a NAS - network attached storage. Those typically are able to hit 30-60 MB/s transfer speeds for the cheap ones, 80-120 MB/s for the expensive ones. I...