Is it possible to eat in such a way that you'll never have to fart?

In short, no. To explain: you have a bunch of beneficial bacteria living inside your digestive tract. These little guys help you digest your food, and in return they get a free meal. This basically means that the food you eat ferments into poo. Fermentation always results in gas being released (think how the plastic wrap on a bowl of dough can puff up as it proves.) So to sum it up, unless you drank a strict water-only diet, anything you eat can give you gas. (Even then, you would probably still naturally swallow some air with the water, and some would make it to the other end as a fart.)