Is it possible to have an AR(0) for streamflow?
Autoregressive models are mathematical models, and like all models, are based on a conceptual model of the system. The conceptual model is that prior flow is a parameter in prediction of current flow. So it doesn't really make sense to have a model where it does not do so. 

Furthermore it is difficult to conceive of a natural system where prior flow is not a predictor. This is where I disagree with Gordon's answer. Even in small catchments, prior flow is a good surrogate for the physical conditions that contribute to flow. The conditions include precipitation history, soil moisture, filling of depressions so flow can accumulate, capacity for trees to intercept precipitation (are the leaves already wet), water-table stream connectivity, etc.

If you don't see a relationship to prior flow in small catchments, that may just mean your daily timestep is too large. I care about the flow variation in very small catchments and monitor the flow on 30-minute intervals.