If a person makes a story up and doesn't copyright it can anyone copy it and then copyright it themselves?
I think that the answer to that is no.
According to 17 USC 102:
Copyright protection subsists, in accordance with this title, in original works of authorship fixed in any tangible medium of expression
And according to 17 USC 101:
A work is “fixed” in a tangible medium of expression when its embodiment in a copy or phonorecord, by or under the authority of the author, is sufficiently permanent or stable to permit it to be perceived, reproduced, or otherwise communicated for a period of more than transitory duration.
Copyright only exists for "fixed" works, and it is not considered "fixed" unless that fixing is done "by or under the authority of the author". It would seem, then, that if it is embodied in a copy without the authorization of the author, no copyright protection would yet exist.
Copyright exists the instant the “story” is placed in “tangible” form
Tangible form means written on paper or on a computer or recorded etc. In some jurisdictions, tangible form isn’t needed and a spoken piece has copyright protection even if it’s never recorded.
It’s been that way virtually worldwide for at least 30 years since the US adopted that method in 1978.
As Dale & DM say, since 1978, all you have to do to get copyright in the US is to "fix" the work in any "tangible" medium of expression.
However, there are certain valuable rights that you only get if you register your work with the Copyright Office. These rights include:
Being able to sue people who violate your copyright. To file an suit for infringing on the copyright to your US work, you have to register the work.
Registration gives prima facie evidence that your copyright is valid along with any facts stated in the certificate.
If you register your work before someone violates your copyright, or within three months of publishing your work, you are eligible for statutory damages, attorneys’ fees, and costs.
If you register your work, you can establish a record with the U.S. Customs and Border Protection for protection against the importation of infringing copies.
The basics of copyright are covered in this flyer, from the Copyright Office.