We lived without a refrigerator for 17 years, from a few years before our sons were born until they were in highschool, 1973 until 1990.
We milked goats every day, had fresh eggs and a large garden, lots of canned stuff,
rabbits and chickens.....it can be done
When we butchered a goat it was for a party. A rabbit or
chicken could be eaten in one meal by the four of us...no leftovers.
In the nineties we lived with public utilities so had a full sized refrigerator, and we kept it full with five of us in the house (my mom lived with us then).
Now, for the past several years we have a very small refrigerator...that is all we need. It was fifty dollars from a student moving out of a dorm near here...a GE that holds plenty for the two of us and even has a fairly large freezer for it's size. Seems like even in a tiny house one could be squeezed in somewhere.