What kind of fish? Some fish basically don't move unless there's food, some never stop moving.
Spacial needs are based, first, on water quality issues. However, if you put a 3-foot-square chain link cage in a river and kept a 2-foot bass in it, the bass would have very clean water but would be extremely unhappy.
Unless it's a fish that mostly sits on the bottom, I think a general rule is to never put a fish in a tank that's less than 5X by 2X, where X is the length of the fish. Very active fish like shiners should probably have more like a 10X by 5X tank. Inactive fish could go in more of a 3X by 2X, assuming their inactivity is constant as opposed to just them being asleep.
Also, a fish in an aquarium should be observed for signs of stress. If they exhibit stress behavior such as unusually high aggression or nonstop pacing, they may need a larger aquarium.
A lot of the minimum tank sizes are probably based on observation, as well. People see a fish looking unhappy in X sized tank, so they move it to Y, then to Z. The fish displays no stress in tank Z, but seems unhappy in X and Y, so Z is considered more or less the minimum tank size.
Basically, if you aren't sure how much space a fish needs, you go and ask people who've kept it. If nobody knows, try a tank and see if the fish looks stressed or cramped. In young fish, keep a close eye out for any signs of stunting.
Filtration isn't as much of a factor as you'd think. Biological filtration is a major component, as that's what handles ammonia, but mechanical filtration (the filters themselves) don't factor in too much. Mechanical filters have three purposes: They hold/oxygenate media so that BB (beneficial bacteria) can grow, they provide circulation/oxygenation, and they help remove physical dirt from the water before it becomes ammonia. Heavy filtration helps when you keep large fish, but it's not required, and it doesn't automatically make a tank able to handle big stuff.