"Infertile"; "fruitless"? How would you describe such a woman in an informal talk to your friend?
I wouldn't use a particular word, I'd just say, "She can't have a baby."
I would never say something like, "She is infertile/fruitless/barren/sterile." – I find that extremely rude.
Historically, a woman with such a condition was considered to be barren, but that word is not used much today (although it is nevertheless likely to be understood). Nowadays, one would probably say she is "seeing a fertility counselor" or some such circumlocution, and people would infer the rest.
I have always used the word barren, although that word is a bit antiquated.
For some, the word barren may carry a negative connotation. For instance, when the word is used in the Bible it's written as if it was a curse. There are Biblical stories of women who were barren but then were blessed by God and were able to conceive. Consequently, someone who is Jewish/Christian/Muslim might find that particular word a bit insensitive, as it seemingly implies they are cursed.
Perhaps the best way to avoid any potential embarrassment is to state the facts plainly, as in, "She is unable to conceive." Even though most people will understand your intent, I'm not a fan of saying, "She cannot have a baby," because that can be taken literally to mean that she is not fit to mother. Just because someone cannot conceive does not mean they cannot have a child through alternative means, such as adoption or surrogacy.
The joke was not meant to provide answers, but a funny lesson on the complexity of the English language - not that other languages aren't complex. Many lessons are best illustrated through humor - although the underlying lessons of that message can be missed entirely. Thanks for the go-around here; I was able to give my daughter a lesson in the fun and folly of language.
I personally would say "she was not able to have children", and if that is too difficult just don't bring it up.
I liked the "euphemism treadmill" concept - very illustrative of the situation - I often get dinged for using the words ignorant, or deception as being negative although I find few ways of expressing the same meaning, and both can be positive or neutral when used that way...