The "web" that is mentioned in the passage refers to the burial shroud woven by Penelope as a way of delaying her need to chose a new husband among the many suitors.
Explanation:
Homer's epic narrative "The Odyssey" tells the story of how Odysseus had been gone from his home for about two decades. His wife had been pestered continuously by the men in the kingdom of Ithaca to remarry and make a new king, for her husband Odysseus had been gone for a long time.
But the cunning Penelope made a plea that she be allowed to weave a burial shroud for her father in law Leartes, who's grown old. She agreed to marry one of the suitors when she finished the shroud. But through cunning and bravery, she would weave the shroud during the day but undo it at night so that the weaving cannot be ever finished until she is sure that Odysseus is back or confirmed dead. This is the web that the passage from Book II mentions, the web being the shroud that she had been weaving as an excuse to not remarry.
