Olivet is a city in Osage County, Kansas, United States. As of the 2020 census, the population of the city was 73. It is located near the south …Olivet is a city in Osage County, Kansas, United States. As of the 2020 census, the population of the city was 73. It is located near the south side of Melvern Lake. Olivet was founded in 1869. By the end of its first year, the town contained a general store, a hotel, a sawmill, and a wagon and blacksmith shop. Olivet was incorporated as a city in 1870.