Thanks to Teslinhiker for the search. I found the following -- "four meals" apparently is a modification of "three meals," the origin of which is not clear. But it does seem to be a popular saying rather than the conclusion of any modern research effort. Also, "three meals" seems to mean food rather than a measure of time; i.e., if you don't get your three square meals a day, everyday, then you will eventually revolt against a system that cannot guarantee your food security.

https://en.wikiquote.org/wiki/Wikiquote:Reference_desk/Archive/2#A_society_is_only_three_meals_away_from_anarchy

I don't know how MI5 got the four meals from. The Times seems to think this sort of rhetoric has something to do with justifying MI5's budget. I'm unable to read to the end of the article without subscription. But now I see that it makes no sense to imagine 1.5 days without food and society will break down completely, with utter panic and chaos as a result. As some posters point out above, societies have survived famines, war, sieges, etc. without falling into chaos. I'd think that at one point in history, going without food for a day or two happened regularly. All of us probably have gone without food for that long or longer.