It depends what you mean by "doomed". In the grandest of senses, yes -- at some point in the distant future, the sun will engulf the inner planets, charring the earth into a giant ball of fire. Presumably, we will have found a way to leave the planet and live somewhere else by then but, much further into the future of the universe, everything will be swallowed up by black holes -- and even those black holes will ultimately evaporate into a cold bath of particles, flitting through space. Our best understanding now is that, eventually, the universe will be a cold, dark and lonely place. It is hard to imagine how life will exist in that realm.
But I do not find that depressing. I find it thrilling and gratifying that we little human beings on earth, through the power of thought, observation and calculation, can make statements about what things will be like in a trillion years. That, to me, is exciting and erases any fear of doom.