Consider a bathtub filled with water, with a plug in the drain. how long would this be considered a closed system?

For a closed system, you need two things:
1) a conservation of mass within the boundaries of the system
2) the ability to freely exchange energy to & from the "closed" system with a surrounding external system

So, the answer is never, since your defining the "system" as the water within the bathtub, and an open bathtub is exposed to evaporation, which is not conserving mass within the defined "system".

Thermohaline circulation and gyres