It's not an entirely lost cause, but there are some caveats.
I work in an environment with a similar ratio. ~400 users and 6 IT people. 2 administrators, 2 techs, 1 on help desk, and 1 web guy. We inherited several kludgy systems when we came on the scene 10 years ago, and daily network interruptions were common and over a hundred trouble calls a day were the norm.
We started by streamlining the networking services. We began replacing the desktop hardware with new equipment to take advantage of warranties and reduce our stockpile of obsolete replacement gear. We upgraded and standardized everything. We purchased tools so we could do a lot of things remotely. We locked down our users. No unauthorized software gets installed, ever. They use a standardized desktop and nothing gets saved to the desktop. Within 2 years we were down to less than a dozen calls per week. It's gone back up since then, but those are external factors.
It was all possible because we had a budget, and really good management.