Aeinautae or Aeinautai were magistrates at Miletus

Miletus

Miletus was an ancient Greek city on the western coast of Anatolia , near the mouth of the Maeander River in ancient Caria...

 around 600 BC

6th century BC

The 6th century BC started the first day of 600 BC and ended the last day of 501 BC.Pāṇini, in India, composed a grammar for Sanskrit, in this century or slightly later...

, consisting of the chief men in the state, who obtained the supreme power on the deposition of the tyrants, Thoas and Damasenor. Whenever they wished to deliberate on important matters, they embarked on board ship (hence their name), put out at a distance from land, and did not return to shore till they had transacted their business.

The historic source is Plutarch

Plutarch

Plutarch then named, on his becoming a Roman citizen, Lucius Mestrius Plutarchus , c. 46 – 120 AD, was a Greek historian, biographer, essayist, and Middle Platonist known primarily for his Parallel Lives and Moralia...

 Moralia

Moralia

The Moralia of the 1st-century Greek scholar Plutarch of Chaeronea is an eclectic collection of 78 essays and transcribed speeches. They give an insight into Roman and Greek life, but often are also fascinating timeless observations in their own right...

 Vol. IV, fasc. 21, Quaestiones Graecae , 32.298c-d:

Translation:Who are the Perpetual Sailors among the Milesians?

When the despots associated with Thoas and Damasenor had been overthrown, two political parties came into control of the city, one of which was called Plutis, the other Cheiromacha. When, accordingly, the men of influence gained the upper hand and brought matters into the control of their party, they used to deliberate about matters of the greatest importance by embarking in their ships and putting out to a considerable distance from the land. But when they had come to a final decision, they sailed back; and because of this they acquired the appellation of Perpetual Sailors.

Further there are three known stone inscriptions from the island Euboea

Euboea

Euboea is the second largest Greek island in area and population, after Crete. The narrow Euripus Strait separates it from Boeotia in mainland Greece. In general outline it is a long and narrow, seahorse-shaped island; it is about long, and varies in breadth from to...

 which feature the word aeinautai. The first one, IG

Inscriptiones Graecae

The Inscriptiones Graecae , is an academic project originally begun by the Prussian Academy of Science, and today continued by its successor organisation, the Berlin-Brandenburgische Akademie der Wissenschaften...

 XII.9.923 from Chalkis, is broken so badly that we are left merely with a list of names and the word aeinautai. A second inscription LSAG 88.21a.S433 from Eretria

Eretria

Erétria was a polis in Ancient Greece, located on the western coast of the island of Euboea, south of Chalcis, facing the coast of Attica across the narrow Euboean Gulf. Eretria was an important Greek polis in the 6th/5th century BC. However, it lost its importance already in antiquity...

, records a dedication of a herma

Herma

A Herma, commonly in English herm is a sculpture with a head, and perhaps a torso, above a plain, usually squared lower section, on which male genitals may also be carved at the appropriate height...

 made by the "association" (koinon

Koinon

Koinon , meaning "common" and interpreted as "commonwealth", "league" or "federation" were a number of associations of cities in ancient and early modern Greek history. The various federations termed Koinon were:...

) of the aeinautai. It dates to the fifth century BC. The third, IG XII.9.909 is a dedicatory inscription from the third century BC, also from Chalkis.