Q: How much space should my chickens have in their run?
A: The "run" is a fenced or enclosed yard space you give your flock outdoors, where they can forage, sunbathe and generally engage in all the chicken-y behaviors that make them happy.
The more space your flock has, the better! With lots of space, your flock will be less stressed. A large outdoor run area can also help balance a small coop. When your flock is concentrated into a very small area, they have more of a tendency to pick on one another and squabble. A larger RUN helps this situation far more than a larger COOP, because it's the run area that gives your chickens the opportunity to express their instinctual behaviors like foraging. Ideally, you want enough space so that the run remains grassy and they can eat bits of weeds and grass, and look for bugs and other tasty things.
Let your chickens out!
Outdoors, the most important thing is that there be something to forage. If your space is small enough to be eaten to the dirt, you might consider providing kitchen scraps, or tossing sunflower seeds or corn into the bedding so your flock can have fun looking for the pieces. A small coop is nice in cold winter areas in the sense that the flock's body heat keeps it warmer than a large coop. But in that situation, they WILL need more space outdoors in the run, and a cover---even a partial cover---will allow your flock to go out and forage in inclement weather and relieve the tension of being too closely confined.
As a rule of thumb, the minimum amount of space you should provide each chicken is 10 square feet, but as we mentioned before, the more space, the better! In fair weather, chickens really don't want to be inside for any length of time, so the indoor space can be small so long as the run provides plenty of outdoor space.
Chickens in mobile, "tractor style" coops can often get along with less than 10 sq feet per bird because the coop and run are moved every day so they always have access to fresh pasture. That keeps them engaged and interested in foraging behaviors. They are ALWAYS interested in being able to access fresh greens, and if their outdoor space is too small, they will quickly forage their run to dust. That creates more work for you, because you'll have to make extra effort to provide good nutrition and keep them entertained so they're not too irritable with each other. A mobile coop solves this problem by simply allowing you to get them on fresh, grassy pasture every day so they don't get bored.
But in permanently sited coop-run setups, the very minimum amount of space they should have is 10 square feet per bird. And in that situation, be sure to get docile breeds that bear confinement well. Some breeds prefer to range far, and can be aggressive with one another when they don't have adequate space.
Before you purchase (or design/build) your coop and run, you'll want to read about the four ways to provide ranging options for your flock, and determine which one is right for you. You may want different features for your coop based on how you decide to manage your small flock of pet chickens!