You definitely can. It's a difficult thing to learn a language by yourself, but if you have the right resources - textbooks, software like Rosetta Stone or Living Language, online practice, etc. - it's absolutely possible. It may take a lot of time and a LOT of motivation, but you can.
I strongly suggest
Language Exchange Community - Practice Foreign Languages for one of your online resources. It allows you to chat and speak in English for the purpose of learning English. Also, watching TV shows and movies in English is a great way to see people speak so that you can get a good idea of different gestures - a more physical part of learning a language.