Success is nothing more than having achieved a goal or desire.
With each success we have we build confidence and self-esteem. Success is like beauty, it's in the eye of the beholder.
In other words each of us defines success in our own way.
Once man stops desiring he stops living.
Since success is about fulfilling a desire I would be inclined to say it's very important to experience a certain measure of success to lead a happy life. It makes you a "winner" in life.
This could range from having a healthy & happy family to owning your own multimillion dollar company.
By the same token if we NEVER accomplish (any of our goals) in life we are bound to feel like "losers". Consistant losing leads to depression and long term depression which is unhealthy.