Gove City is located at 38°57'34N, 100°29'14W (38.959576, -100.487316). According to the United States Census Bureau, the city has a total area of 1.0 km² (0.4 mi²), all land.
As of the census of 2000, there were 105 people, 45 households, and 27 families residing in the city. The population density was 106.7/km² (279.3/mi²). There were 57 housing units at an average density of 57.9/km² (151.6/mi²). The racial makeup of the city was 98.10% White, 0.95% Native American, and 0.95% from two or more races. Hispanic or Latino of any race were 1.90% of the population.
There were 45 households out of which 28.9% had children under the age of 18 living with them, 51.1% were married couples living together, 8.9% had a female householder with no husband present, and 40.0% were non-families. 35.6% of all households were made up of individuals and 22.2% had someone living alone who was 65 years of age or older. The average household size was 2.33 and the average family size was 3.15.
In the city the population was spread out with 30.5% under the age of 18, 4.8% from 18 to 24, 23.8% from 25 to 44, 18.1% from 45 to 64, and 22.9% who were 65 years of age or older. The median age was 40 years. For every 100 females there were 75.0 males. For every 100 females age 18 and over, there were 69.8 males.
The median income for a household in the city was $23,438, and the median income for a family was $42,813. Males had a median income of $22,500 versus $20,625 for females. The per capita income for the city was $11,870. There were 20.0% of families and 34.5% of the population living below the poverty line, including 58.5% of under eighteens and 7.7% of those over 64.