I find the answer rather unsatisfying as it relies on the fact that pi is about 22/7, not any of the other properties that make pi such an important number. I guess the "within 0.002" should have been a clue, but I still think there must be answer that relies on the fact that the containers are circles. In that spirit, I present the following problem, which is, I guess you could say, a sequel to the original.
You have just completed RaptorInsurance's problem, and as a reward, he or she has provided you with a shiny new protractor. However, while you were admiring its beautiful design, an evil demon came along and stole the 3L container from the original problem, as well as kidnapping RaptorInsurance. Thus you are now left with the same bowl and 4L container as before, and a protractor. The task now is to end up with exactly pi liters of water. Your solution may not depend on the fact that pi is very close to some particular fraction, and the only reason for anything less than perfect precision is minor errors in measurement, pouring, etc. not in the method.
I have an outline of how I would do this, but not a complete solution. By the way, as has been pointed out, you could get arbitrarily close by using the ternary expansion of pi, but I'm not granting you infinite time. After all, you already have infinite water, so don't be so greedy! Besides, I think that would count as repeated applications of the already-forbidden close-to-a-certain-fraction rule.
Now that I think about it, the physics is beyond me, but I think you could also get the answer if you put the 4L cylinder into a centrifuge, and you calculated the exact RPM necessary to get 4-pi L to spill out over the edges. Of course, this assumes you have access to information such as the viscosity of water, and so on. Anyway, that's just silly.