There are certain situations in life and in careers where you may only get one shot to achieve a particular goal. However, most situations we experience are not life and death situations. We do get second chances, and the trajectory of success is not always straight up.
To overcome the fear of failure you must not give up at the first sign of failure. If you do give up, you will only be further conditioning your fear of failing. That does not mean you should do the same things and expect different results, but you can change based on what you learn from failure.
For example, say a particular individual is interviewing at a law firm with the dream of becoming a great attorney. Suppose the individual is rejected and does not get the job. If the person looks at the failed interview as an indicator that he or she will not be a great lawyer, then the fear of further rejection will inhibit future success. The person could very well learn from the experience, change his or her approach to interviewing, and secure an opportunity at a different law firm.
While persistence will not overcome core talent and limitations in skills, giving up due to the fear of failure will not allow individuals to achieve their full potential.