Is it normal to strive for an unachievable goal?

Yep.
Perfectly normal.
Me, i'm planning to live forever.
So far, so good.

completly normal, especially for high acheivers and unrealistic/slightly realistic goal setters like myself.
This is great and will keep you self motivated and give you drive to acheive more than the normal person, just dont let these goals be any harm to yourself and recognize that if you do not reach these goals it is nothing to beat yourself up about or be depressed about for too long as you realize the danger f failing right now. On the other hand do not let this fact of expected failure allow you to quit or give up on any of your goals.
Very normal for competitve alpha's

not that anyone cares but I did keep trying and everything turned out amazingly. Much faster and unexpectedly than i ever thought was possible. It was everything I ever wanted, perfect. :)

Meh, you got something else you'd rather be doing? If not, then why not?

Yeah i agree with dappled. I think striving for things that you may think are unattainable keep you going through life. It is good to set very high goals. My mom always said to shoot for the moon and even if you miss you will be among the stars.

Keep trying. People give up much, much, much too easily. A translation of the Buddha's last words are "strive on with diligence". I agree with him. A life spent not trying is not a life.

why would you strive for an impossible goal? If you think you will never achieve it, why would you pursue it? If you accept failiure, you will fail. Dont give up, and you shall succeed.

If there is a specific example that might help. I'd say as long as working toward that goal doesn't prevent you from leading a successful life, and as long as you're ok with the possibilty of never reaching the goal then it's fine. For example, if you want to win a marathon and you run every day and practice but never win, there are still physical and mental benifits from working at that. However, if you want to be a rock star and you focus all of your time and energy on that and don't hold down a realistic job and make money and pay bills that's probably not a good idea. It all depends on how it affects your life and how you can balance your life to achieve the goal.