The expression does not refer to a specific type of fee related to astronomy or space exploration. It's simply a way to express the feeling that a fee is unreasonably expensive.
Here are some examples of how the term might be used:
* "They charged me an astronomical fee just to fix a minor scratch on my car."
* "The hotel's room rates were astronomical during the holidays."
* "The consultant's fees were astronomical, but his expertise was invaluable."
It's important to remember that "astronomical fee" is a subjective term. What is considered astronomical to one person may not be to another.