Probably in an anthology. Judging by age of local library collections, probably well over 40 years old. The story had a future rural or agricultural setting possibly in an extraterrestrial colony of Earth. At least one of the characters had something called a "barton"; whether this was a creature or an object I do not remember, but "barton" was definitely the word used. With a bit of Googling I discovered that there is a poem by Thomas Hardy in which the word barton means barnyard but this may not have been the meaning in the story. I do not remember any other science fiction story that used the word barton as a type or category of a thing rather than the name of a particular character. (As a contrary example I seem to remember a movie in which the lead character called his dog "Dog".) I would have been likely to browse through anthologies containing authors like Asimov, Blish, Bradbury, Clarke, Pohl, etc.