And though they were built for utilitarian purposes, barns became objects of beauty to artists and architects.

European architects of the early 20th century became enamored with the spare construction of American barns, said Sandy Isenstadt, a professor of modern architecture at Yale. “To them, barns and silos represented exactly the sort of strict functional thinking, without concern for aesthetics, that a new architecture might be founded upon,” he said. “So they saw such buildings as beautiful  a beauty that descends from attention to function rather than from concern for appearance.”

Wendell Minor, an artist who lives in Washington, Conn., and has used many of Connecticut’s barns in his illustrations  including one in a United States Postal Service postcard series  compares barns to lighthouses. “They have the same strong visual elements,” he said. “People are nostalgic for them in the same way they are for lighthouses.”

Most people recognize that barns are worth saving because they give the state part of its appeal, preservationists say.

Kent Gilyard, 68, a barn restorer who owns two 19th-century barns in Litchfield, said there is something about them that just makes people feel good. “Maybe it reminds them of a simpler time, even though farming wasn’t simple,” he said.

The grant project, funded through the Connecticut General Assembly, is intended to help property owners understand that “barns and outbuildings do not necessarily have to be torn down,” said Mr. Levine.

Image Samuel Averill outside the milk room of the barn he uses for his apple orchard. Credit... Wendy Carlson for The New York Times

Mr. Averill, a ninth-generation farmer, operates his apple orchard on land that his ancestors bought from the Indians in 1746. Even though he spent his childhood milking cows and building forts out of hay bales, he said, he sometimes wonders whether building a new barn would not be easier than fixing an old one. One side of his barn buckles out, possibly the result of a shifting stone foundation, he said.