As a Greek guy, mandatory military service = doing various chores and jobs for the Greek army for a year for free, so maybe you chose well. Wait until you are 45, pay the fine and you will be fine.
I would happily pay a fine. When I was looking (7-8 years ago) I do not recall there being an option for this. I am 40 now so would happily look at doing this in 5 years. Thanks for the tip