Finiteness of the Algebraic Closure
The Artin-Schreier theorem asserts that these are precisely the real closed fields, which roughly speaking are the fields which behave like $\mathbb{R}$, and that their algebraic closures have degree $2$ and are given by adjoining a square root of $-1$. The Wikipedia article gives several examples; the simplest one is probably the real algebraic numbers $\mathbb{R} \cap \overline{\mathbb{Q}}$.