Urwumpe's translation was pretty accurate. I am not aware of a "δμαίμον" word. Propably meant to write "όμαίμον" which is the word "ομόαιμον" in Attic dialect (homoemon) meaning the same blood.
The phrase from Herodotus Histories, book 8 (Urania),144 is this:
το ελληνικόν εόν όμαιμόν τε και ομόγλωσσον και θεών ιδρύματα κοινά και θυσίαι ήθεά τε ομότροπα
There is also the Greek nation of the same blood and language and same sanctuaries and common sucrifaces and morals.
(Ancient Greek to Greek translation Aggelos Vlahos "Galaxias" 1971, Greek to English translation - mine).
It has been used as a nationalist phrase by many over the years, and interpreted by some, that Herodotus used it to define who are Greek and who are left out.
To know what it was actually about you need some context. Before the battle of Plataea, the Spartans were worried that the Athenians would make some kind of deal with the much stronger Persians, so they sent emissaries to Athens to prevent that.
The meeting took place at the Ecclesia of Demos in Athens and the Athenians answered that they would not make any kind of deal with the Persians and that they would remain allied with the rest of the southern Greeks. That phrase is (according to Herodotus) part of the answer the Athenians gave to the Spartans.
Here is the
link to the original Greek text of Urania, you can find the whole answer the Athenians gave at 144.
Here is an English
translation