Prefab" tab at top of website) dating back to the 1920s, it was moved to Princeton from New York or upstate New Jersey by J.P. Whiton-Stuart and Mary Marshall Ogden in the 1930s. They later sold the house to the Veblens, who also owned the 1875 farm cottage nearby. After Elizabeth Veblen's death in 1974, the house was donated to Mercer County with the expressed desire that it become a nature center, museum and library. The county then rented it out until 1998, at which point it was boarded up.
The "hay barrack" serving as a woodshed in the background, torn down in 2008 or so by the county, was a very unusual structure originally designed for the roof to rise and fall to accommodate hay as it was stacked. You can see how the four corner posts extend up from the roof, which slides up and down on them. (Note: Some internet research since writing this post shows that the hay barrack design originated in Holland to store an overflow of hay from the main barn. The footprint of an old barn is just behind the hay barrack in the photo.)
In 2001, a local architectural firm determined that the "Veblen House and Cottage are eligible for listing as an historic district, for their association with the nationally significant mathematician and scholar Oswald Veblen." That listing was not made, but attests to the historical importance of the buildings.