I am looking for a hypernym for all places that provide food, beverages, or both, of any quality or price — such as restaurant, cafe, fast-food and so on.
The name of the industry in the US is “food service” (or “foodservice”); according to Wikipedia the corresponding term in BE is “catering industry”. Within the industry, individual outlets are “food service operations”.
In ordinary speech the most ‘generic’ term is “restaurant”, which embraces both ‘sit-down’ (table service) establishments and fast-food outlets, but tends to exclude institutional operations such as hospital or corporate cafeterias. “Eatery” is sometimes used, but it’s old-fashioned and implies a down-scale establishment; referring to a Michelin-starred restaurant as an “eatery” would be distinctly jocular.
Travel guides, newspapers and the like will often use verbal headings: “Eating”, “Eating Out”, “Dining Out”, and so forth.
tchrist asks for a term embracing hot-dog stands and the like. Within the industry they're still “food service operations”; they may be grouped with outlets like Burger King as “quickservice operations”. In ordinary conversation—well, they’re not “restaurants”. They are “eateries”, but as I said that’s not a term used much any more. I don’t think there’s any generic term embracing everything from the hot-dog stand to Maxim’s. If I had to find an X to plug into “What sort of X do you want to have lunch at?” I’d have to resort to ‘place’ or something equally vague.
In law the category is often referred to as a "retail food establishment", as opposed to a wholesaler or caterer. Hot dog stands and Michelin-rated restaurants are both included.