In the events of Starbound, it is attacked and destroyed by the Ruin during the intro mission. It was home to the Terrene Protectorate, an organization with the goal of exploring and offering aid across the galaxy. An emergency broadcast was sent to other Humans after the destruction. As shown in the Off-Earth Human Census and its Report, few humans live away from the Earth in comparison to the amount on the planet. Earth also appears to be the only planet with arcologies, other than Decayed biomes where the arcologies appear destroyed.
An Earth Model is sometimes sold at the Frögg Furnishing store. Other Solar System planets can be bought here too, with racial descriptions containing additional lore about life in the Solar system before Earth's destruction.
You cannot travel to Earth or any other real planets in Starbound, except Earth briefly during the intro mission.
In the background of the intro mission, as pictured, one can make out the London bridge, an obvious easter egg to London, where CF operates. It also establishes that the Protectorate academy is in London as well