Yeah, you could. I've built cabinets with the bare minimums. I'm gonna toot my own horn here, but my easiest cabinet was the Vigolix I designed. All straight cuts, but has a very sharp design. Although I did round some corners by sanding them.
Here is my project linkDrChek and Slappyhooper both made Vigolix cabinets as well. DrCheks is a lot like mine (but nicer), and Slappyhooper made a cool 2 player full size version of the cab.
I think the absolute bare minimum you need to make a cabinet is a saw, a drill with a hole bit that PL1 Mentioned, screws (will work better for you than nails), sandpaper, paint (you can buy spraypaint or a roller/brush and a bucket of paint), bolts for mounting joystick and monitor.
But anything you can borrow that is a power tool will help.