How much water should I drink to flush my system of barium?

Probably not enough. Just water by itself is not likely to evacuate barium from the GI tract. You will probably need a mild laxative to help ensure its removal.
Barium Sulfate.... ..Used in the ba enema or ba swallow is not water soluble, and is not absorbed into the body. Fiber and water will flush it out better than water alone.