Seeing as I work every day with individual DNA molecules which behave discretely (as in, one goes into a cell or one doesn't), and on the way to my advisor I walk past a machine that determines the 3D molecular structure of proteins... yeah.
This edifice not being true would rely on truly convoluted laws of the universe that emulate it in minute detail under every circumstance I can think of, but not doing so under some circumstance not yet seen. I am not sure how to quantify that, but I would certainly never plan for it being the case. >99.9? Most of the 0.1% comes from the possibility that I am intensely stupid and do not realize it, not thinking that it could be wrong within the framework of what is already known. Though at that scale the numbers are really hard to calibrate.
I think a more plausible scenario for the atomic theory being wrong would be that the scientific community -- and possibly the scientific method -- is somehow fundamentally borked up.
Humans have come up with -- and become strongly confident in -- vast, highly detailed, completely nowhere-remotely-near-true theories before, and it's pretty hard to tell from the inside whether you're the one who won the epistemic lottery. They all think they have excellent reasons for believing they're right.
How much confidence do you place in the scientific theory that ordinary matter is made of discrete units, or 'atoms', as opposed to being infinitely divisible?
More than 50%? 90%? 99%? 99.9%? 99.99%? 99.999%? More? If so, how much more? (If describing your answer in percentages is cumbersome, then feel free to use the logarithmic scale of decibans, where 10 decibans corresponds to 90% confidence, 20 to 99%, 30 to 99.9%, etc.)
This question freely acknowledges that there are aspects of physics which the atomic theory does not directly cover, such as conditions of extremely high energy. This question is primarily concerned with that portion of physics in which the atomic theory makes testable predictions.
This question also freely acknowledges that its current phrasing and presentation may not be the best possible to elicit answers from the LessWrong community, and will be happy to accept suggestions for improvement.
Edit: By 'atomic theory', this question refers to the century-plus-old theory. A reasonably accurate rewording is: "Do you believe 'H2O' is a meaningful description of water?".