I am looking to store the activity of an user, but I am not sure where to store it. I dont think database is a option as it will be very big than. I am looking to know as to how
You might want to look at django-activity-stream which is an implementation of the activity streams spec. This stores a list of actions in the database and allows the following of users/entities to give something similar to Facebook if this is what you are interested in.
However, as you mention, this can end up with enormous sets of data which might be a bit much for a conventional single database approach. I'm not sure how sites like Twitter deal with it but unless you plan to scale up very quickly the standard database approach would probably last you a while.