#metropole

The Lawrence Arms - Beautiful Things

The city was sweating in the summer heat I wrote melodies of regret to a def and empty street I was so lonely it started to rain The lightning and thunder were singing my name I thought those songs could save me I thought those songs could save me

Don't kill all the beautiful things I was searching for truth, in the dust of my days I was so lost, and I was so young

Great question!

To answer "How would the food supply chain work for city states," I would answer by answering your other question "Would they control agricultural land, or would they have to buy food from nearby (or perhaps not so nearby) agricultural regions?" by saying: it's both.

Paradigmatically, a city-state consists of a metropole (the city in question) and the periphery (subordinated lands and territories under the city's control). (Or metropolis and hinterland, if you prefer). To further break this down, the periphery consists of a mix of different lands, including both overseas colonies and tributaries, conquered (and thus lesser and weaker) city-states and their peripheries, and the agricultural hinterland around the metropole. To take just this last category, we can think of the relationship between Athens and Attica, Rome and Latium, Venice and the Mestre and Veneto (everyone always focuses on Venice's overseas empire, but it's rarely remembered that Venice at its height basically controlled all of northeastern Italy and fought Milan for northcentral and parts of northwestern Italy), or Florence and (eventually all of) Tuscany.

How far that agricultural hinterland extended beyond the city's walls depended a lot on transportation technology. Brett Devereaux has very good (but lengthy) explanations of the difficulties of overland grain trade here and here, but the TLDR is that, as a rough rule of thumb, "the price of grain doubles every hundred miles it is moved overland." Those kind of price increases aren't really affordable, so I think if you're looking for a rough rule of thumb, a hundred miles is probably a good maximum radius for an agricultural hinterland, whereas the minimum radius is probably around 7-12 miles (based on medieval urban regulations for agricultural markets), which was roughly how far a cart could travel in a day.

However, certain factors can change the effective radius of the hinterland:

The more and better roads you have, carts can move faster and come from more directions/places, which effectively expands that cart/day radius.

If the city is on a river (and this is one of the main reasons why most historic cities were on rivers, if they weren't on coasts), you can use river-barges to transport grain. Sail-barges could travel 14 miles an hour in good winds, and tow- or pole-barges could do 10-40 miles a day (depending on whether they were going upriver or downriver). Moreover, because of buoyancy, barges can carry much heavier loads than carts, which makes them much more eficient in transporting bulk goods like grain.

Finally, the population density and degree of urbanization matters, because it raises the possibility of making partial trips, because smaller population centers will act as local metropoles and more efficiently bring in grain from rural areas allowing for more efficient routes; also, higher density and urbanization allows for the creation of a network of granaries that allow you to store grain along the way so that you can make partial trips rather than covering the whole distance from the city to the very edge of the periphery.

However, the hinterland usually wasn't enough on its own to supply the city-state, but the same advantages of wind-speed and buoyancy also meant that a long-distance overseas grain trade was absolutely viable in both the Ancient and Medieval worlds. So for example, Greek city-states would draw on grain from southern France, southern Italy and Sicily, the Black Sea, and Anatolia; much of Rome's grain supply came from North Africa and Egypt; and so on. Moreover, as Fernand Braudel points out, the interconnected grain trade of the Mediterranean was the indispensable foundation for southern European urbanism from the Middle Ages through to the Early Modern period, and the geospatial dynamics of that system did not really change that much until the invention of the steam engine.

To answer your final question - "What measures would they take in times of war to avoid having their food supply cut off?" - it was usually by a mix of metropolis/hinterland logistics and long-distance trade. A well-organized city-state would have granaries set up both in the city and in the more urbanized areas of the periphery and would have contingency plans to harvest and transport as much grain as possible to the city to use as reserves in a time of crisis. (You'll note that this leaves the periphery in a really bad situation in times of crisis, and this prioritization of the metropole over the periphery is one of the main reasons why it sucked to be in the periphery, which is why there was so much competition over who got to be the metropole and why there were so many rebellions on the peirphery.)