Hansen was born in Zenith, Wyoming. He attended the public schools of Jackson, Wyoming, where he has lived for most of his life. He graduated from the University of Wyoming in 1934, and later served on the University's board of trustees from 1946 to 1966, and as its president from 1954 to 1962.

Hansen was a cattle rancher and became involved in several agricultural/cattlemen's associations and local politics. He was elected governor of Wyoming in 1962 and served in that position from 1963 to 1967. He did not run for a second term in 1966, instead choosing to run for the United States Senate seat from Wyoming which was being vacated by retiring Republican Milward Simpson. Hansen won the election, receiving 51f the vote and defeating Democrat Teno Roncalio. He was reelected in a landslide in 1972, defeating Democrat Michael Vinich with 71f the vote. During his time as Senator, Hansen was known for being socially and fiscally conservative and was one of the eight Senators to vote against the Equal Rights Amendment. Hansen retired from the Senate in 1978, declining to run for a third term. He moved back to Jackson, Wyoming, where he still lives, and chaired several financial and civic associations.

Upon the death of former Senator Hiram Fong of Hawaii in August of 2004, Hansen became the oldest living person to have served in the United States Senate.