I'm not a sports photographer, but I've shot a lot of stuff at night in football stadiums (marching bands). It really depends on the lighting in the stadium. With really good lighting, you'll be able to get good shutter speeds at f/4, and the 300 f/4 would do very well.
If your stadiums are dimly lit, you may need f/2.8 or even faster (higher ISO) to get your shutter speed up. Unfortunately on your budget, you can't do 300mm and f/2.8. You could give up 100mm and get a 70-200 f/2.8 (or a 200 f/2.8L), but that may not be enough reach. If you can pony up even MORE money, consider the Sigma 120-300 f/2.8. (still significantly more than the 300 f/4, but much less than the 300 f/2.8).
For most high school stadiums, f/4 really won't be enough, even at ISO 1600. (of course, f/2.8 has issues here too). I was getting shutter speeds at a decently lit high school stadium in the 1/80s range for ISO 1600 f/4. Of course, at f/2.8, it's still a bit slow, though at f/2.8 and ISO 3200 (with a 30D or 40D), you could get 1/320 at f/2.8, which would be enough to stop a lot of action (but not all).
College / pro stadiums, or very well lit high school stadiums, you might be able to swing f/4 and still get action shots for sports.