Eurysthenes was a son of Aristodemus and Argia, daughter of Autesion, and had a twin brother, Procles. He married Lathria, daughter of Thersander, King of Kleonae, who was the sister of his sister-in-law Anaxandra. By Lathria he was the father of his successor, Agis I, the founder of the Agiad dynasty of the Kings of Sparta.

The title of archēgetēs ("founding magistrate") was explicitly denied to Eurysthenes and Procles by the later Spartan government on the grounds that they were not founders of a state but were maintained in their offices by parties of foreigners. Instead the honor was granted to their son and grandson, for which reason the two royal lines were called the Agiads and the Eurypontids.

The story of the double kingship of Sparta begins with the invasion of the Peloponnesus by the Dorians and their Aetolian allies under three Heraclid commanders, Temenus, Cresphontes and Aristodemus, the three sons of Aristomachus. According to the account assembled by Karl Otfried Müller from the classical authors, the state of Elis (allied with the Aetolians) provided a guide, Oxylus, for passage through Arcadia after the army crossed the Gulf of Corinth from Naupactus. Arcadia gave them a central point from which to attack anywhere else in the Peloponnesus. Their presence was contested by a united Peloponnesian Achaean army (except for Arcadia) under Tisamenus, an Atreid. The Achaeans lost and were commanded to evacuate to Athens, though many did not, and much of the region remained unconquered.

The three commanders nevertheless divided the Peloponnesus they did not yet fully possess, and, following the signs of the gods, Aristodemus received Sparta. Whether he was ever actually in possession there is disputed: one tradition holds that he was, making him the first king of Sparta, while a second asserts that he died beforehand and that the Dorians brought his infant twin sons to Sparta as kings under a regent. Aristodemus was assassinated at Delphi by the Atreids, having had no time to designate a successor, and the mother did not know which twin was the elder.

The oracle at Delphi resolved the problem by suggesting that both sons be made kings, which is the origin of the dual monarchy. Theras, Argeia's brother, was made regent. Since it was still necessary to designate the elder, they chose the one the mother fed and cleaned first, Eurysthenes.

The untimely death of Aristodemus, with other events, has served as a basis for dating the reigns of the first ten kings of Sparta in the line known as the Agiad. The Return of the Heracleidae, the closest event to a Dorian invasion available in legend, must coincide with the entry of Aristodemus and his brethren into Arcadia, which, based on the chronology of Eratosthenes, happened 328 years before the generally accepted date of the first year of the first Olympiad, 776 BC. Eratosthenes's date is therefore 1104 BC, which is taken as the year of Aristodemus's activity in Arcadia, his fatherhood and his assassination. Eurysthenes was therefore born in 1104 BC, the first year of his reign if the regency of Theras is discounted.

Pausanias states that the end of the First Messenian War fell in the first year of the 14th Olympiad, a date of 724/723 BC if the first Olympiad was 776/775 BC. Kings Polydorus of the Agiads and Theopompus of the Eurypontids were reigning at that time, roughly in mid-reign, and the end of the war must fall 379 years after the return of the Heraclids. According to Isaac Newton, also a classical scholar, the ten kings reigned an average of 38 years each, which can be used as an estimator. By this reckoning Eurysthenes would have ruled in 1104–1066 BC, with an unknown margin of error, as much of the data is relatively uncertain.