Need Help building a fallout shelter in house
As you are apparently in a rural location, you may not need a permit, however the only way to know is to check with the local building inspector.
In answer to the question can it be done, absolutely, however as noted by ARI001, this can be a complex job because it involves cutting load bearing elements. Also, a staircase is a surprisingly complex item to fabricate from scratch, because of clearance issues, support of the base, rise and run issues, and handrail requirements.
Now on to the fundamental question. You stated that you want to build a fallout shelter. I am old enough to remember a period of time when fallout shelters were popular. A fallout shelter was designed to protect the occupants against radioactive fallout from an atomic war for a period of time, perhaps 30 or 60 days, after which presumably the occupants would emerge onto a charred wasteland and resume living a long and happy life. I don't think you are actually intending to build such a shelter, I get the idea you actually want a storm room, which is a radically different type of structure.
For example, you need an air filtration system for a fallout shelter, not necessary for a storm room. You need a large supply of canned goods and drinking water for a fallout shelter, not necessary for a storm room. You need blast protection for a fallout shelter, not necessary for a storm room. You need emergency electrical generation capability for a fallout shelter, probably not warranted for a storm room. You get the idea.
So if you are actually interested in building a fallout shelter, let us know, I can steer you to web sites that discuss issues associated with such shelters. If it is a storm shelter, let us know, that is a much more easily achieved goal.