In the anime All Might tells in a fight half of his respiratory organs were destroyed and he lost his complete stomach. How can All Might live without his stomach?
Living without a stomach isn't as impossible and absurd as it first sounds as removing the entire stomach is currently one of the more common treatments for stomach cancer too.
You can find a short explanation how part of the small intestine turns into something like a pouch to take over the role of the stomach in this video, while there are numerous articles written by doctors, patients and their relatives on the internet.
Just linking the first few hits Google dropped for me: