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The answer is actually pretty straightforward. If it's just Bunning, the Dems can keep calling for unanimous consent and Bunning has to stay there on the floor to block it. As soon as he needs to sleep or just go to the bathroom they can rush to the floor, call for unanimous consent and then that's it.
Add a few more people, though, or add two or three dozen then it gets very easy to fight off. You just have one guy take an hour shift, followed by another person, followed by another. And maybe you have two on the floor at all times to prevent any mistakes.
If you've got 40 people, it's not hard to do it in shifts and even have it be relatively painless.
And remember, you don't have to keep talking. All you have to do is sit their and be ready to object.