A First Clarification on Anxiety

Is there any room for worry over things that are truly frightening?

The fact that the Bible calls anxiety a sin does not mean that it doesn't also acknowledge that there are frightening things. Peter himself, in 1 Peter 3:6, tells us not to "fear anything that is frightening". And Paul tells us that if we take our anxieties to God in prayer, "the peace of God which surpasses all understanding will guard [our] hearts and minds in Christ Jesus" (Phil. 4:7). If that peace surpasses all understanding, that means that the human mind cannot understand how peace is possible in a given circumstance; i.e., there are "legitimate" reasons to be terrified. So yes, there are frightening things in this fallen world: car crashes, war, homicide, fire, cancer, heart attacks, theft, tornadoes, freak accidents, kidnappings, and the list goes on. But while there are scary things in this world, we need to remind ourselves that we serve a God who is control of every one of those possibilities and all the others I haven't listed, that we are His children under His tender care, and that He only does what is good for us and for His glory. And so we fight to take even these kinds of anxiety to our Father in heaven. 

In the context of speaking to his disciples about the persecution they will face as His servants (certainly a little frightening), Jesus says this:

"So have no fear of them, for nothing is covered that will not be revealed, or hidden that will not be known. What I tell you in the dark, say in the light, and what you hear whispered, proclaim on the housetops. And do not fear those who kill the body but cannot kill the soul. Rather fear him who can destroy both soul and body in hell. Are not two sparrows sold for a penny? And not one of them will fall to the ground apart from your Father. But even the hairs of your head are all numbered. Fear not, therefore; you are of more value than many sparrows." (Matthew 10:26-31)