First, we need to understand that the belief in a literal Hell is not ubiquitous among religions. Even within Christianity, roughly 30% of Christian Americans don't believe in a literal Hell according to a 2007 Pew Forum on Religion & Public Life / U.S. Religious Landscape Survey. Also, the answer to this question is very different if we are talking about the non-religious who never believed in Hell and the non-religious who used to believe in Hell. For those non-religious who never believed in Hell, there is simply no fear, worry, or anxiety about it, just like they probably don't fear the possibility of being eaten alive by carnivorous space unicorns. So rather than "ex/non-religious people" let's look at atheists (non-believers) who used to believe in Hell.
It's important to understand that those who do/did believe in Hell, have very different ideas of what Hell is and how one ends up there. Below is an excerpt from the chapter "What the Hell?" in my book The Concept.
Hindu scriptures, which have been around long before Judaism and even longer before Christianity, describes 'a dark world, filled with evildoers and their relentless cries of pain and agony, undergoing different kinds of torture and punishment as a consequence of their bad deeds in their previous lives.'
But what about Judaism, which serves as the foundation of Christianity? Most modern translations of the Old Testament (the Hebrew Bible) do not even contain the word “Hell”. If you are familiar with the King James Version of the Bible, you might recall that “Hell” is mentioned over a dozen times. This is a mistranslation of “the grave” that has been corrected in more recent translations. This “grave” (the Hebrew word “Sheol”) is a physical location beneath the earth where physical bodies go when dead. But in Numbers 16:30+33, we see that the living can go there as well — kind of like visiting New Jersey. While this “grave” place is no Disneyland, it’s a place where the dead appear to “lie silent” (Psalm 31:17) rather than scream in eternal torture. Most importantly, there is absolutely no mention of people being sent to this “grave” because of what they do or don’t believe.
In the New Testament, the idea of Hell becomes even more complicated. Hell is no longer a place for bodies, but a place for ghost-like images of the person. It is now a place of torment and anguish (Luke 16:23-24). This idea of Hell, usually referred to as “Gehenna” or “Hades”, is a major theme in the New Testament as it’s mentioned well over 100 times, about 70 times by Jesus himself. But Gehenna, often mistranslated as Hell, was really just a valley nearby Jerusalem that had a bad history. Yet Revelation does suggest a place of eternal torture. Confused yet?
I have heard many Christians on the more liberal side refer to Hell as simply a "separation from God," as they find it difficult to reconcile an all-loving God with a God that would allow us to be eternally tortured. Clearly, "separation from God" is not as scary as "a dark world, filled with evildoers and their relentless cries of pain and agony, undergoing different kinds of torture and punishment." The point is, mild or non-violent beliefs about Hell tend not to lead to a fear of Hell post-belief.
How one believed they might end up in Hell is also a factor in how much fear one might still carry about Hell post-belief. For example, Calvinists believe that Hell is predetermined for some people, and there is nothing these unfortunate people can do about avoiding it. Some believe it depends solely on one's beliefs (e.g., believing that Jesus is God to avoid the Christian Hell - or not believing that Jesus is God if you want to avoid the Muslim Hell), some believe that it depends on one's actions, and some believe it is a combination of both. So for example, if a former believer in Hell also believed that Hell was a result of one's actions or deeds, and one believes that he or she is a good person, they would have no reason to be fearful of Hell. This is sort of like driving within the speed limit even if you don't think any cops are around, and not worrying about getting a ticket.
Another factor in how likely an atheist is to fear Hell and to what degree is their level of confidence in atheism. To use the Dawkin's scale of 0-7, 7 being 100% confident that no gods exist, those who are higher on that scale are less likely to fear the possibility of Hell, whereas those who have recently left their religion would be more likely to have residual fears of Hell. Overall, although I know of no surveys that have polled atheists with this question, as an active member of the secular community, I would say that fear of Hell among members in this community is not at all common.
Now, to your question about overcoming the fear. To be afraid of something one does not believe is pretty much the definition of irrational. While one may claim non-belief from a rational perspective, one may still have a feeling-based or affective residual belief as a result of years of conditioning that may take quite a bit of time to be consistent with one's rational belief. In the meantime, my suggestion to mitigate any residual fear of Hell would be to learn as much as you can about the origins of the idea of Hell, the widely different views of Hell among believers (especially Christian preacher Rob Bell), and ask yourself the same theological question billions of believers have been asking for centuries: could a god who loves me really allow me to be tortured for eternity?
While there are atheists who have some residual fear of Hell due to their religious upbringing, most don't fear Hell any more than they fear the wrath of Zeus. As one rationally considers the origins of Hell, the theological and philosophical problems with the idea of Hell, and the power such an idea has to control behavior, it becomes abundantly clear that Hell is not a concept worthy of our fear. The more we understand this by reason, the more we will feel it, as well.
By the sweat of your brow you will eat your food until you return to the ground, since from it you were taken; for dust you are and to dust you will return. - Genesis 3:19
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