The Enterprise crew receives an automated message from the enigmatic Sheliak: Remove the humans on Tau Cygni V in four days. However, this is not a simple warning or directive - the Sheliak are a non-humanoid species with little regard for human life, and would exterminate any humans found in their path. In fact, only due to a treaty with the Federation did they feel compelled to notify their intention to colonize before taking further action.
There is no record of a Federation colony ship being sent there as it contains levels of hyperonic radiation lethal to humans, which doesn't explain why the Sheliak indicate otherwise. The Enterprise arrives in the system to find what looks to be a small colony on the surface. Lt. Cmdr. Data, the only crewmember unaffected, takes a shuttlecraft down to the planet to coordinate the evacuation. Once he arrives, he finds that the sensors were incorrect, it is a colony of 15,000 people - the descendants of a wayward colony ship. The colonists' ancestors found a means to survive within the radiation, but initially suffered heavy loss of life before an effective defense was found. Although it would normally be a simple matter of beaming the colonists off the planet, hyperonic radiation renders the transporters useless.
With time running out, Capt. Picard takes the Enterprise on a course to intercept the Sheliak colony ship on route. The Sheliak have no interest in compromising: they feel that the treaty clearly states it is within their rights to colonize Tau Cygni V. Picard is forced to determine a legal basis within the 500,000 word treaty that might stall the Sheliaks' plans. Meanwhile on the planet, Data cannot convince the colony leader, Gosheven, that their presence is in violation of a treaty only temporarily protecting them from destruction and that evacuation is the only option. The colony leader holds that the sacrifices made by the original colonists allow them the right to live on the planet.
Ultimately, Picard finds a solution within the treaty - the Sheliak can either choose to wait three weeks for the transport ships to evacuate the colony, or take the matter to arbitration by the Grisellas, a species currently hibernating for the next six months. The Sheliak leader relents and the ship turns back. Meanwhile Data is able to convince the colonists that the colony is not worth dying for and that other challenges await them on a new world.