Hi Suiyer,
Are you planning to stay in one spot for the whole month? Most people would probably choose to travel around, but sure, you can also spend time in one spot and really get to know it. Berlin I am sure is a great place to stay for four weeks without ever getting bored. You could spend a week or so exploring the city, and then move on to the outskirts and neighboring places like Potsdam. You could easily take day trips by train from Berlin, for instance to Dresden or Wittenberg, and you could do all that without ever even renting a car.
In four weeks, you might even get to know a few people. I am not a great hostel person, so I really can't give you much advice on that, but they say that hostels are great for meeting other people, especially when you are traveling solo. As far as safety goes, they are not going to mug you in a hostel, but just use common sense and don't leave your valuables lying around unattended.
Yes, we do have Airbnb, but a word of caution, especially for Berlin: Make sure your accomodation is properly registered. Airbnb has a bit of a bad reputation here because it competes with regular apartment rental, raising rental prices enormously. I myself had an experience staying in a small rental apartment for a weekend which I believed to be part of a hotel, and a neighbor came knocking on the door, letting me know that this was really illegal. Not a desirable way to get to meet locals.
Let me add that the month you have chosen is not necessarily the nicest month in terms of weather. November can be grey and cold and rainy. But then, you just have to go by the German saying: There is no bad weather, there is only bad clothing. :-)