How many slots does it have at the moment? You could add/upgrade to bigger drives to fill all the 3.5" bays it has, use the 5.25 bays for 3.5" hard drives too with converter brackets,add a PCI card so it can take more than 4 drives. All this for hardly any cost other than the internal hardrives you would need to buy anyway for the time being.
This will allow you to start organizing/getting your videos on drives straight way. If and when the computer finally gives up the ghost, instead of replacing it, put all the hard drives in something like these. A much cheaper option than buying another computer. Plug these into any computers already on your network and add as you go if the need be.
A basic 4 bay enclosure
With this next one, if and when you fill 4 hard drives, instead of buying another full 4 bay enclosure, you could buy the individual pull out cradles with each extra hard drive and just swap them when needed. This way it gets cheaper as you go.
4 bay swappable enclosure
Or what the hell, have this and be done with it. Still cheaper than another, even barebones computer I would think. And if you can fill all 8 x 300GB hard drives, then you really need to stay in more
8 bay swappable enclosure