Eurycleia is an old lady who is the "faithful old housekeeper" of Odysseus's household. She does not want Telemachus to carry through with his plan to get a ship and go off in search of his father.
The reason that she gives for not wanting him to go has to do with the suitors. She says that once Telemachus leaves, there will be no one to oppose the suitors. They will then be able to do whatever they want and take whatever they want. Besides, she says, there is no way that Odysseus is still alive. So she opposes Telemachus going because there is no point in it and it will just let the suitors have their way.
Eurycleia is also thinking about Odysseus at this point. She thinks of Telemachus, whom she helped raise from infancy, as a young Odysseus, whom she also cared for as a child. Their lives are parallel to her. So, when Telemachus raises the idea of going off to sea to search for his father, Eurycleia can't help but think about what happened when Odysseus went off to sea twenty years ago... he never came back. She is afraid that the same thing will happen to her precious little Telemachus, that he too will never come back to Ithaca. She believes that Odysseus is long dead at this point, which makes her really sad, so she doesn't want the same fate to befall Odysseus's son. Even though she thinks he will never come back, she is still extremely loyal to him. This means she cares a lot about Telemachus too, and want to protect him from anything that might happen out at sea. She is also afraid that if Telemachus does come back, the suitors will be so angry that he left, that they will harm him or kill him.