Can one become fluent in a language without studying it formally?

First of all, we need to define what "study" means. A 1/2 year old is certainly studying the world, the movements our mouths make to pronounce words and sounds. The baby doesn't have access to grammar books, but learns by observing and repeating. Babies learn that grammar exist and overregulate what they know - they say "horse" and "horses", "sheep" and "sheeps". The advantage adults have is that we don't have to sit and wait until those around us say the words we want to learn - we can ask them for it! We can even buy books and study materials to accelerate our learning. It is possible to learn a language without ever opening a book about it, but you need to dedicate some time to learn it. You need to learn words and sentence structure by listening or reading. You need to produce the language and have your errors corrected. Look up Benny Lewis' method on fluentinthreemonths. If this is time spent joking around with friends and enjoying yourself, it might not feel like studying.