Can cats and chickens co-exist in a small yard? Would a chicken climb over a 4 foot chain length fence?

You be the judge!

Chicken and cat
YouTube - Cat and chicken
Or
YouTube - Chicken doesnt like kitty

And I would say yes about the fence, unless you clip the wings!
YouTube - Stupid farmer...(watch the chicken in the background)

Depends on the cat, some of them will get a chicken. We had cats and chickens growing up and they were fine together, the cats kepy mice and rats out of the coop. 4 feet might not be tall enough, 6-7 feet is better. You could put some mesh on top to keep them in.

Before you go too far down this path, are you in an area that will let you keep chickens? Not all spots will.

In answer to the questions about the chain link fence,
A chicken with or without been clipped can still jump at a hight of around about 8 feet high which means that seriously the chicken will be able to get out if it wants to,
As for cats and chickens if you get a young kitten and let it grow up around your birds you will not have a problem, as long as it grows up around them it will see your birds as safe and not for killing, however it will still attack wild birds or chances of it are high anyway even if it doesn't,
I used to keep cats and birds togather quite happily and I breed birds, my uncle breeds parrots and has cats living in the bird room quite happily, it all depends on how the cat is brought up, if young and placed in with your birds if you keep an eye on it and it shows signs of wanting to attack you can catch it befre it does anything by the scuff of its neck and move and then after this has happened a few times it will learn,
Cats and birds can grow up quite happily togather :)

But seriously I would consider getting a higher fence and if living in areas then make sure it is an area you can keep chickens in,

Forgot to say also that my uncle also keeps chickens running about and the cat is fine with them, the chickens sometimes try and bat the cat away but 99% of the time they don't bother and the at doesn't bother at all, :)