Really? When I google feynman hairy green sphere, I get as the second hit a quote from Surely You're Joking, Mr. Feynman! which runs:

Richard P. Feynman ... Finally they state the theorem, which is some dumb thing about the ball which isn't true for my hairy green ball thing, so I say, “False!

Clicking through reveals the whole story, of course. And the third hit is a blog post which excerpts the key summary:

I had a scheme, which I still use today when somebody is explaining something that I'm trying to understand: I keep making up examples.

For instance, the mathematicians would come in with a terrific theorem, and they're all excited. As they're telling me the conditions of the theorem, I construct something which fits all the conditions. You know, you have a set (one ball)-- disjoint (two balls). Then the balls turn colors, grow hairs, or whatever, in my head as they put more conditions on.

Finally they state the theorem, which is some dumb thing about the ball which isn't true for my hairy green ball thing, so I say "False!" [and] point out my counterexample.

It's the "hairy green ball" from "Surely you are joking...":

For instance, the mathematicians would come in with a terrific theorem, and they're all excited. As they're telling me the conditions of the theorem, I construct something which fits all the conditions. You know, you have a set (one ball) disjoint (two balls). Then the balls turn colors, grow hairs, or whatever, in my head as they put more conditions on. Finally they state the theorem, which is some dumb thing about the ball which isn't true for my hairy green ball thing, so I say, "False!" If it's true, they get all excited, and I let them go on for a while. Then I point out my counterexample.