First, people simply bought fewer perishables. Instead of going shopping once a week and storing everything you bought until you needed it, you'd go to the butcher on the day you were actually planning to make roast beef or whatever. This is why milkmen used to be a thing: people got around the fact that they couldn't keep milk for long by simply having fresh milk delivered to their doorstep every morning. The whole production process moved a lot faster: cattle would be herded right into the middle of the city, slaughtered, butchered, and be in the shop the next day, ready for the cook to put on the table in the evening.

Although salted and dried meat was used, people didn't rely on it so much after tinned food became common in the 19th century. There are dozens of other things you can do to food to make it suitable for storage: pickle vegetables, turn fruit into jam, turn milk into cheese, etc, etc. Knowing the various methods of food preservation was a skills most housewives would have before it became easy to just buy stuff from a store in the 1950s and 1960s. Today we generally don't consider the fact that a lot of the foods we eat were originally just ways of keeping things edible for a long time.

Most houses had a pantry, a cupboard (or whole room) for storing food. Basements worked too, although in a city most people wouldn't be able to afford to give over a whole floor of their house to food storage. Any place that's dark and cool will stretch out the lifespan of food for a few days. However, people weren't entirely without artificial refrigeration. Iceboxes were very common, unless you were very poor. In rural areas peasants could harvest ice in the winter, store it in an insulated box, and have enough left by summer to store food. The wealthy built ice-houses, big underground bunkers, on their country estates which could store several tons. If you lived in a city then there was also a whole industry for ice-harvesting, which supplied massive amounts of ice and was big business as late as the 1930s.

Starvation wasn't even common in the 19th century in Europe, as the railways eliminated the primary cause: poor food transportation networks. There were one or two exceptions to this when particularly bad natural disasters hit (Ireland, looking at you), but certainly by the beginning of the 20th century famine was almost unheard of in most of Europe. However, that's only peacetime. During WW1 the Allies blockaded the Central Powers, and by 1917 Germany and Austria were experiencing severe food shortages. Several hundred thousand excess deaths during WW1 can be attributed to malnutrition.

Although people didn't generally starve to death, higher food prices than America meant malnutrition was more common in Europe. A survey done in Britain during the Boer war (about 1901) showed that a shocking proportion of urban industrial workers didn't meet the height and weight requirement to serve in the army because they had suffered childhood malnutrition. Even by WW2 most Europeans had far fewer calories in their daily diet than Americans, and were on average an inch or two shorter as a result.

In Nashville in the 1940s, I remember an “ice man” delivering a huge cube of ice and putting it into a wooden, porcelain-lined box . Perishables would be kept there. I think shopping was daily or close to it. Grocery stores were close by and small. My mother and we two children would walk to the store and bring home groceries in a large paper bag. Milk was delivered. My brother was a toddler so he and the bag rode in the baby carriage. By the end of that decade refrigerators were common. I am 79.