Old English etan (class V strong verb; past tense æt, past participle eten) "to consume food, devour, consume," from Proto-Germanic *etan (source also of Old Frisian ita, Old Saxon etan, Middle Dutch eten, Dutch eten, Old High German ezzan, German essen, Old Norse eta, Gothic itan), from PIE root *ed- "to eat."
Transferred sense of "corrode, wear away, consume, waste" is from 1550s. Meaning "to preoccupy, engross" (as in what's eating you?) first recorded 1893. Slang sexual sense of "do cunnilingus on" is first recorded 1927. The slang phrase eat one's words "retract, take back what one has uttered" is from 1570s; to eat one's heart out is from 1590s; for eat one's hat, see hat. Eat-in (adj.) in reference to kitchens is from 1955. To eat out "dine away from home" is from 1930.
Siberian Mongolian people, 1580s, from Russian samoyed (11c.), traditionally literally "self-eaters," i.e. "cannibals" (the first element cognate with same, the second with eat), but this might be Russian folk etymology of a native name:
The common Russian etymology of the name Samoyed, meaning "self-eater," deepened the Russians' already exotic image of far-northerners. The most probable linguistic origin of Samoyed, however, is from the Saami — saam-edne, "land of the people" [Andrei V. Golovnev and Gail Osherenko, "Siberian Survival: The Nenets and Their Story," Cornell University, 1999]
Which would make the name a variant of Suomi "Finn." The native name is Nenets. As the name of a type of dog (once used as a working dog in the Arctic) it is attested from 1889.