How do Catholics believe God is eternal if Jesus died on the cross? The doctrine of the Trinity states that there are three Persons: God (the Father), Jesus (the Son), and the Holy Ghost.

The Church also teaches that Jesus died for the sins of humanity.
Jesus is 100% God and 100% man, and at the same time not mixed or separate.

So if Jesus is said to have died for the sins of humanity, and Jesus is God, then God died.

An entity that dies cannot be considered eternal.

Is the dictionary term "eternal" not correct, or are our rational minds incorrect in thinking that an entity that dies is not eternal?  (I don't mean to sound rude or harsh, but this is a theological matter, and the seriousness of the matter sometimes requires stating things bluntly.)

If one just states the usual, "It is a mystery," I think that might be another way of saying it doesn't make sense—just accept it. I am looking for a more direct and specific answer than that, based on Catholic theology.
I'm going to answer with an earthly illustration and then tie the scriptural doctrine to it at the end, so bear with me.

Suppose there is a video game developer.
He spends his days typing the computer code that is the game. 
He experiences the game from the perspective of a programmer. 

He knows how the graphics will be drawn to screen and how all the pieces will interact. 
Then the game developer decides to experience the game differently.
He hits the preview mode button (f12 perhaps) and now when he types on the keyboard a character jumps and moves around. He is experiencing the game world as a player through what is called an avatar. 

Something happens in the game and the avatar 'dies', which is to say that the game code prevents the developer from experiencing the game world through the avatar anymore. 
At this point the game developer can... 

A) Hit ESC and go back to experiencing the game as a programmer, looking at code
B) Wait (or use a shortcut key built for developers) for the avatar to respawn. 


  In either case the developer exists even when the avatar is surrendered or lost.


Now for the Scriptual doctrine tie in...

The Spirit of the Son as 'developer':


  1 Corinthians 8:6 "Jesus Christ, through whom all things came and
  through whom we live."
  
  Hebrews 1:2 "his Son [Jesus]... through whom also he made the
  universe."
  
  John 1:2-3 2 He was with God in the beginning. 3 Through him all things were made; >without him nothing was made that has been made. 


The Spirit of the Son then limited itself to experience the created universe through an avatar:


  Philippians 2:5-7 Christ Jesus, 6 who, although He existed in the form
  of God, did not regard equality with God a thing to be grasped, 7 but
  emptied Himself, taking the form of a bond-servant, and being made in
  the likeness of men.


The Spirit of the Son was able to hit the 'respawn' button, resurrect the human body avatar-


  John 10:18 18 No one takes it from me, but I lay it down of my own
  accord. I have authority to lay it down, and I have authority to take
  it up again. This charge I have received from my Father.
  
  Ephesians1:19-20 according to the working of his great might 20 that
  he worked in Christ when he raised him from the dead


Returning to overseer of the 'source code'-


  Ephesians 4:10
  
  10 He [Jesus] who descended is the one who also ascended far above all
  the heavens, that he might fill all things.
  
  1 Peter 3:22  Who [Jesus] has gone into heaven and is at the right hand of
  God, with angels, authorities, and powers having been subjected to
  him.


The existence of the Son is not threatened by the loss of an avatar. 

Remaining questions

The above is a cheap painting of doctrine that is common among many Christian denominations. Opinions diverge on the details of the Son's relationship to the human avatar.


  Did the Spirit of the Son leave the human avatar when it died and
  re-enter it on resurrection, or did the Son stay bound to the avatar
  and wait for resurrection 'in the dark'?
  
  Did the Son return to whatever the Son was before the incarnation, or
  is there something different about the Son after that experience? In
  other words, does the Son continue to bind itself to a immortalized
  version of the Jesus body?


You can find the second question discussed briefly
HERE