But consider these words in a "non-computer" context.
There is a TV inside a room.
I access the TV - I go inside the room and use the TV.
I retrieve the TV - I go inside the room, pick up the TV, and take it out of the room. There is also a strong implication that I am the one who originally put the TV in the room.
You didn't ask about this but we normally talk about data/records/information stored on a computer not inside it. Elizabeth was able to start her computer but unable to access/retrieve any records stored oninside it.