If you face a gigantic task and feel fear of failure analyzed the situation fearlessly and honestly and figure out what will be the worst that could possibly happen as a result of this failure.
After discovering the worst that could possibly happen, you will find that worst is not as worst as it appears to be. It’s   turning into reality is highly remote.
When you prepared for worst, your fear will be banish.
It is my humble opinion.

I think my "flight" response is disabled somehow. :P

Fear and love are the most powerful feelings, meaning they are the ones that affect your heart the most. The circumstances that trigger fear has changed radically during the last few centuries, but the bodys response to fear is still the same as it has always been. It causes "stress", which is to say it causes the body to expend almost no energy on digestion and in return adds that energy to the arms and legs. People who feel a variant of fear on a constant basis thus have poor digestions and uneasy limbs.
To give an example, if I come home and find an unexpected letter saying I have to pay a large amount of money which I currently do not have, my body reacts as if a predatory animal is trying to kill me, because fear is fear and it triggers the 'flight or fight' response.

Fear manifests in several ways:

Terror - fear for your existence right now. Useful because it makes you run faster or hit harder.

Panic - when fear consumes you and your rational mind is disabled. Useful because it makes you run faster, hit harder and ignore pain. Potentianlly crippling as it can also cause catatonia - mental paralysis, like that of a surprised rabbit. It is said that 98% of people in fact "freeze up" though I personally have a hard time believing this to be true.

Dread - fear for something/someone specific. Useful because it makes you run faster, hit harder and heightens senses around a probable threat.

The kind of fear you can avoid by using the great method posted above is called 'worry' and serves absolutely no purpose.

I remember hearing about a statistic/survey being done that showed that what people fear most are speaking to a crowd. Go figure.

Cheers!

I've faced most of the fears in my life - didn't like tight spaces so I took up potholing - all that stuff so done most of em are they lessened?
Yes, up to a point.
The one fear I can't face is bears!
Illogical - never seen a bear - there are no  bears in England but that's the only one - until I find one I didn't know I had.

I think the distinction between "fear" and "anxiety" that other posters made is a useful one. I also think that the advice to face our anxieties (which we often perceive as fears, since this then justifies our anxious behaviour patterns) and release our fears is the best way to deal with them. Sometimes the hard part is recognizing and acknowledging what is a true fear vs. what is an anxiety.

For years I had a deep-seated phobia of sharks. I would have terrible recurring nightmares. It affected my quality of life: I would wake up in terror, my legs cramped because I had been in the fetal position, yet psychologically I would be unable to stretch out my legs, convinced there was a shark at the end of my bed. Likewise, i would refuse to take baths as a child (only showers), and failed swimming lessons multiple times since I adamantly refused to go into the deep-end of the pool. This was not a true fear, but a crippling anxiety.

Recently I was surprised to remember a dream where there was a shark that was attacking a turtle, whose response was simply to tuck itself into its hard shell so the shark was rendered impotent. When I awoke, I was shocked to realize that I felt no fear or anxiety, either during the dream, or recalling in my waking state. As I've come to analyze my early childhood phobia, I've started to think that the shark symbolizes my own unchecked appetite, something that at times has come close to consuming me as it goes on a rampage of consumption (I have addictive personality I now keep in check). Realizing that what i thought was an external threat is actually a projection of an internal trait was a real breakthrough for me. I am learning to face the parts of myself I'm not happy with (those so-called demons) and recognize them as parts of myself rather than external forces. This has been my way to defeat a long-held "fear."