There are plenty of signs with the phrase "outside food", and even Disney states this in their FAQ:
Can I bring outside food and beverages into Disneyland Park or Disney California Adventure Park?
Is this term acceptable in formal English?
If not, is there anything more concise than "food not purchased from this establishment"?
You ask about "formal" English but it's not quite clear what that means. If you're referring to establishments which don't allow "outside food" and are doing so in a statute or position paper, say, you would probably use some form of periphrasis such as "food carried in by visitors" or "food prepared off-premises by entities not under the control of the establishment".
I can see where “outside food and beverages” could be slightly ambiguous (= picnic food/drinks?) to someone unfamiliar with its common usage in the food/beverage industry.
I’ve always taken “outside food and beverages” to be a shortened, and therefore arguably less formal version of “food and beverages from outside sources” so, depending on your audience and/or purposes, perhaps “No food or beverages from outside sources allowed” would be slightly more suitable.
In the Disney FAQ you linked to, the answer contains the phrase "your own food and beverages." So while "No outside food" is a widely accepted idiom, if you're looking for something more formal, you could go with "Guests cannot bring their own food."
The term is constructed "properly", and the intended meaning does not stretch either of the words -- it definitely is food, and "outside" here has the same sense as in "outside opinion" or "outside help". I don't see any grammatical reason to reject it in formal contexts, unless you reject those too because you disapprove of using "outside" as a modifying adjective when "external" might be preferred.
"Outside food" has idiomatic meaning in a particular trade. If you want it to be taken with this precise meaning, that depends mostly on whether the intended audience understands it.
Without a context that dictates this idiomatic meaning, the word "outside" is somewhat ambiguous. The term could perhaps mean "food that is out-of-doors" and be used in connection with picnics. All you have to worry about is whether or not it's taken to mean specifically "outside" from the point of view of an establishment that sells food.
I think that in most contexts calling for formal language (for example a contract, or in written correspondence with the Queen) you might very well want to assume ignorance of anything so demeaning as the jargon of the catering trade ;-) Therefore don't use it without definition, but that still doesn't mean one cannot use it.
If you're looking for an alternative anyway, then "your own food" is quite common. Maybe it still doesn't pin the meaning down precisely for legal purposes, but it will generally be understood and does not come across as either jargon or slang. For a sign, "food not bought on the premises". Although I suppose some pair of wags will each bring in a sandwich, sell them to each other for a penny, and claim that the food therefore was bought on the premises, just not from you, so they're allowed to eat it!
This idiom is not common in British English, and the confusion mentioned by others is very possible: "outside" might well be read as being to do with whether for that was purchased on the premises can be taken out, for example into a garden or onto the street.
