Question

Why doesn't Eurycleia want Telemachus to leave?

She doesn't think he can manage a ship by himself.
She doesn't want to leave Penelope unprotected.
She doesn't want to leave the palace unprotected.
She is afraid that the suitors will try to kill him.

Answer 1

I believe that Eurycleia doesn't want Telemachus to leave because she doesn't want to leave the palace unprotected.
She believes that if he leaves, the suitors will try to plunder the palace and steal everything, and there will be no one to protect either the palace, or Penelope. 

Answer 2

The best answer is that she is afraid that the suitors will try to kill him. The reason why Eurycleia want Telemachus to leave because she is anxious of what will happen to her or him whenever he leaves - in this case, her suitors would want to kill him.