Looks like a manual. So no, can't drive it without the shaft.
Completely fixable. extension housing can be replaced in the car. Buy a donor transmission (can even be blown up) for $100 or less and a "new" driveshaft, spend a few hours crawling around in gear oil, and it'll be fixed. I definitely would, but I just put a brand-new rear subframe in a rusty '00 Outback with 325k miles on it, so I might not be the best person to ask.