So I have my own computer with 250 gigs of storage, and the rest of my family shares another one, also with 250 gigs. My dad just bought a couple of 500 gig backup drives, one for each computer, but I was wondering if there was a way for us to share one drive.

I wirelessly connect to the modem that the other computer has a wired connection to, but I don't think we have any other sort of network besides that. I apologize, I don't know much about this aspect of computers.

So, can we share a drive with what we have? Or should we just use a separate one?

You should be able to set up a home network through the modem/router, although my own attempts at this were less than successful.
Run the network set up wizard on each computer and set the wrokgroup name the same on both computers when running the wizards, you can find this at the task panel in network connections which is a sub menu in control panel.

Then right click the hard drive you wish to share on the network in my computer and select properties then the sharing tab, and select I understand the risks of shring link, which will give you the menu that allow you to click a tick box to enable the drive to be shared on the network.
MY gear
Vintage v100 les paul.
POS Ion strat
Vox ad50vt
15 watt practice amp
Ibanez rg570