It is also a fact that there is no inherent right to feel safe, nor is there an inherent right to feel anything.
Well, if we have the right to life and liberty, I think it is reasonable to argue that we have the right to be safe--only to a degree, of course, because as you said complete safety is unachievable. The questions are the means and who decides how, as well as what is real safety versus merely fooling oneself into feeling safe. In a free country, the people should have the right to defend themselves using the most effective tools--this comes from the virtue of being a living creature, born of nature. It provides the most possible real safety as opposed to merely feeling safe, and it promotes the most liberty and self-determination, which should be the American way, while the other way that makes some people "feel safe" is the opposite, working against liberty, which is un-American.
But let's say there were, and that I don't feel safe in "gun-free" zones. Since I have a right to enter them and in many cases am even required to enter them, they should be eliminated because they violate my inherent right to feel safe.
That's a good point. While I argued above that we had some right to BE safe, through a combination of sensible laws (e.g. murder being illegal), effective law enforcement, and allowing the people to arm themselves as they individually deem fit, there is no right to FEEL safe--that is, you're free to feel any way you want, but there is no guarantee of how you'll feel. If the means are available to let you BE safe (as much as possible) as well as free, then that's all that matters. If a person irrationally fears guns and the idea of an armed populace--besides criminals who are armed anyway--then they're free to seek help, as well as take classes in reasoning and logic, but they shouldn't try to take away the fundamental, inalienable natural rights of the people.
You can easily demonstrate how utterly silly this notion of a right to feel safe is. Somebody above mentioned fear of clowns. There are lots of people who are afraid of the dark, of dogs, of closed places, of open places, of water, of bridges, etc. And we have to admit, people have died in situations involving all of these things (except maybe clowns). So they can feel safe, we must create an environment in which people can live their lives without ever encountering a clown, the dark, a dog, an open space, a bridge...
See how silly that is?
The human mind largely works on association, and it can be quite difficult for some people to logically separate certain things and ideas from one another when necessary. For instance, guns are, quite frankly, designed to kill living things, and killing people is generally considered a bad thing. This association, for some people, is virtually unbreakable--guns kill, killing is bad, so guns are bad, period. It doesn't matter how much logic you apply to the issue, this association will stubbornly persist nonetheless, especially after people transfer their fear of others to guns, as described above.
While the example you just gave may sound silly to a person who can reason to the point of breaking strong associations like this when presented with the facts, it's not so silly to others when GUNS specifically are involved. In short, they're fixated on guns as murder machines.