What is the only thing people call their own and who stays true to their beliefs in "1984"?

I suspect that at the deepest level of "own," there is nothing the people can call their own.  When you consider that the state can take away your ability to say "2 + 2 = 4" and can make you truly turn on the love of your life, what else can you 'have' except trivial "things," and even these are in short supply.  

And I would hazard a guess that there is no such thing as a "belief" in "1984."  However, it all depends on how you define "belief."  You can believe in Santa Claus for a while --- and many people believe in things that the rest of us may not (like predicting the future, ghosts, etc).  When I read your question, I tried to think of something that anyone in the story believed it, and came up empty, unless you say that O'Brien believes that power is the only ultimate, and that using power is the only value.  Perhaps we should be talking about values and not beliefs, although these too are in short supply.