How can I make use of this non-standard network cabinet in my garage?
This is one of the residential structured media cabinets that you can get at any of the big box stores, made by Leviton and several others (your pic is too low resolution for me to read the name on the phone block, there may also be a name on the cabinet door). 

You can buy additional blocks of all sorts of things to fit in these, such as network patches, switches, routers, etc.



It's probably much cheaper just to buy standard 19" stuff from Monoprice, and then regular 19" rackmount switches/routes from manufacturers that actually specialize in network gear (as opposed to light switches and plugs), but that's just my opinion. I used standard stuff at my house, it was fairly cheap and looks great -- I'll post a pic later. 

It really just depends on how you want it to look when completed, how much work you want to do vs using what's there, and of course how much you want to spend.