Are there number fields based on ultraradicals? So you can create quadratic fields, cubic fields, and quartic fields by just taking the nth root of some integer, and some are even unique factorization domains or principle ideal domains, like $\Bbb Q$[$\sqrt{2}$]
Now these are based on the solutions to polynomial equations. Are there number fields you can construct like quadratic or higher that are based on solutions to polynomial equations that don't have closed form solutions (quintics or higher)? Are any of these unique factorization domains or principle ideal domains?
 A: In your question you're confusing a number field with its ring of integers. It's $\mathbb{Z}[\sqrt{2}]$ that's a principal ideal domain. 

Are there number fields you can construct like quadratic or higher that are based on solutions to polynomial equations that don't have closed form solutions (quintics or higher)?

Yes. For any irreducible polynomial $f(x)$ with rational coefficients you can construct a number field $\mathbb{Q}[x]/f(x)$, and by the primitive element theorem every number field arises in this way. For example, by Eisenstein's criterion $x^5 + 3x + 3$ is such a polynomial even though there is no quintic formula. 

Are any of these unique factorization domains or principle ideal domains?

Yes. For example, for any $n$ there is a cyclotomic number field $\mathbb{Q}[\zeta_n]$ where $\zeta_n = e^{ \frac{2 \pi i}{n} }$ is a primitive $n^{th}$ root of unity. This can also be presented as $\mathbb{Q}[x]/\Phi_n(x)$ where $\Phi_n(x)$ is the cyclotomic polynomial of order $n$. Many of the class numbers of these number fields (by which I mean the class numbers of their rings of integers) are known; in particular, $\mathbb{Q}[\zeta_7]$, which has degree $6$, has class number $1$. It's known that the ring of integers of $\mathbb{Q}[\zeta_n]$ is $\mathbb{Z}[\zeta_n]$, so $\mathbb{Z}[\zeta_7]$ is a UFD. See also this list of number fields with class number $1$. 
There is a nice fake proof of Fermat's last theorem you can write down where you implicitly assume that $\mathbb{Z}[\zeta_n]$ is always a UFD, and the story goes that this is the proof Fermat had in mind, but this turns out to be false in general; the smallest counterexample is $n = 23$. 
You can learn all this and much more by picking up any textbook on algebraic number theory. 
