The linguistics uses the term language acquisition to describe the process of becoming fluent in a given language. This term is selected to address the ambiguity of language studying or language learning.
Can one become fluent in a language without (formally) studying it?
Of course. Every baby picks up the language of their parents and/or environment he is being brought up in. Many children until certain age are able to pick up languages in the same manner as toddlers do that, without requiring any textbooks or dictionaries. Even some adults do, as there are always exceptions.
When I was about 9 years old I went to a small town where my grandparents lived. It turned out that the majority of this town were Tatars, an folk of Turkish origin, who used predominantly Tatar language for communication. As a native speaker of Russian I could understand only some words (3-4%), which were borrowed from Russian.
It was a nightmare in the beginning and I felt embarrassed every time I used a wrong word or said something wrong that made everyone (well, children) laugh at me. Ultimately I become fluent in it just by listening to my grandparents, watching some TV with them and talking to people around me. But I can't read or write in this language, and if I need to read something, I first read that aloud and only then I can understand it.
What is the requirements to become proficient in a foreign language?
The most crucial requirement is exposure. You need to be exposed to the language, it should be a part of your environment, your daily routine.
Second is involvement. I have friends who've been living in countries like Finland, Denmark, Greece and still not speaking it beyond 2-3 common phrases and 20-30 everyday's words. If you want to learn it, you must need this language for doing something, e.g. be (deeply) involved with the language.
Third is reference. In order to learn the language you need to grasp the reference between what is said/written and the actual context. How you grasp this reference: either using a dictionary/grammar book/memorizing some rules (the academic way), or inferring its from the context/people's reaction/friend's explanation -- it is up to you.
Computer games in language acquisition
I have never studied English. My foreign language at school was French, I spent 6 years at school diligently learning it, being jealous of my friends who went to another school offering English as foreign language (in Soviet Union where I went to school the language you learn was determined solely by the district school you are assigned to, no choice was given to pupils whatsoever).
When I was around 12 my uncle purchased me a home PC, a clone of the popular Sinclair Spectrum 48K. It had some nice games completely in English, including several adventure games, like "LOTR: The Shadows of Mordor", "A Worm in Paradise", "The Hobbit".
Unlike today, back in those times adventure meant that you have to communicate in (simplified) written language with the game, giving commands like "go north", "examine room", "take knife", "cut rope with knife", "pick stick" etc. No mouse, no clicking over object, just hardcore English. I remember being stuck in one place while playing "The Shadows of Mordor". The game implied that there might be a hole under the tree stump, but how to get to there? I literally spent days trying to figure out what I need. Ultimately I tried "over stump with sword" -- oops, my sword breaks in halves, but the stump is uprooted and I finally can proceed to the next location (turning out to be an underground maze). So, I learned a new word, which, alas, I've never used since then.
Time changes, today there is no need (unless you are really interested in this genre -- as I am!) to play those kind of games. Start playing any games pretending to be open-world (like Terraria or Minecraft), keep reading all the articles in Civilopedia while playing Civilization. You'll soon find yourself knowing and understanding many everyday's word, as well as some not really much useful, like "potion", which still comes to my mind first when I feel thirsty.