Is it possible for people to come out of depression and anxiety on their own?

Yes: Especially for mild or moderate levels of depression and anxiety many do get past it. If they learned valuable self-knowledge during the time, they may even avoid relapse. Others do not have a good outcome - some become more troubled and add problems. You can try to improve your quality of life and watch how it goes. Then maybe add psychotherapy and see. Only add pills if you must! Good luck!