"Barren eyes" is not a special well-known phrase or idiom - the words mean what they say, but figuratively since the context is artistic--poems are a work of art by definition.
Eyes are considered the window to the soul/spirit, and "empty eyes" would mean one's eyes look like they are drained of soul or spirit. The fact that "barren" also is often used to talk about a piece of land's inability to grow vegetation is likely an additional meaning leveraged in the metaphor - i.e. "the eyes look like nothing positive can grow within."
Looking at the meanings of words in poems in this level of detail is an interesting way to fully understand a poem and appreciate the depth of the impression that the writer was trying to create.