Q:

What is this "Nor"?

And I saw Tityos, son of glorious Gaea, lying on the ground. Over nine roods he stretched, and two vultures sat, one on either side, and tore his liver, plunging their beaks into his bowels, nor could he beat them off with his hands.
— The Odyssey

This nor is not like not . . . either.
This nor is similar to never could he beat them off. . . .
Is this correct? Is this an archaic use?

A:

It may be a little archaic, but it's perfectly comprehensible.  Unlike the other respondents, I do not believe an implied previous phrase is required; it is simply and he could not beat them off... together with the rule that `and not =nor' (and possibly a reluctance to have too many ands in a sentence).
The Oxford English Dictionary agrees:

nor
...
5. Following upon an affirmative clause, or in continuative narration, with the force of neither or and…not.
1523 Ld. Berners Froiss. I. cxxxv. 162, I greatly desyre to se the kynge my maister, nor I wyll lye but one nyght in a place, tyll I come there.
a1578 Lindesay (Pitscottie) Chron. Scot. (S.T.S.) I. 26 To mak hir purgatione that scho was frie of all misrewlle...nor gave na counsall thairto.
1631 May tr. Barclay's Mirr. Mindes i. 39 The whole coast is most sweetly verdant, ...nor hardly, is there ground any where more abundantly fruitfull.
1667 Milton P.L. iii. 626 A golden tiar Circl'd his Head, nor less his Locks behind...Lay waving round.
1697 Dryden Virg. Georg. iii. 161 His Age and Courage weigh: Nor those alone.
1738 Johnson London 260 Then shall thy friend, nor thou refuse his aid, ...forsake his Cambrian shade.
1788 Trifler No. 22. 291 The little creature cried and laid down, nor could all our beating raise it.
1821 Byron Heaven & Earth iii. 673 Away! nor weep!
1871 R. Ellis tr. Catullus lxi. 205 Come nor tarry to greet her.
1875 Jowett Plato (ed. 2) I. 423 Nor among the friends of Socrates must the jailer be forgotten.

A:

This way of using nor is indeed somewhat archaic.
Normally, nor requires that another negative has been employed somewhere previously. In your Odyssey quote, for example, we might imagine something like this:

Two vultures sat, one on either side, and tore his liver, plunging their beaks into his bowels. He could not rise himself, nor could he beat them off with his hands.

However, in older texts (especially, in my experience, older translations of Greek texts), nor could sometimes ‘piggyback’ not only on an expressed negative, but also on anything that implies a negative somewhere in the situation.
In this case, the despondent impuissance of Tityos creates such a situation. There is no explicit negative, but it is easy to imagine that the situation as a whole functions as a kind of ‘abstract negative’ that nor can be used to emphasise. Simply using not would just make it a plain statement that these birds were tearing out his innards and he couldn’t get rid of them—but using nor emphasises that the birds did not stop, he did not like it, it was not good … the whole thing was, basically, negative.

