When does an inner product induce a norm?

An ordered field where $a^2+b^2$ is always a square is called a Pythagorean field. As you observe, not every ordered field is Pythagorean, but each ordered field has a Pythagorean extension. If you really want $L^2$-norms you could always extend your ground field to a Pythagorean extension field.

As you noted, to construct an an inner-product space we require the basefield $F$ of the vector space $V$ to be a quadratically closed subfield of $\mathbb{R}$ or $\mathbb{C}$ (i.e. every element of $F$ must have a square root in $F$).

However, a norm is a function $n:V\rightarrow [0,+ \infty)$, as opposed to an inner-product which is map $\langle \cdot ,\cdot \rangle :V\times V\to F$. In the case of the basefield $\mathbb{Q}$ therefore, we cannot construct an inner-product space from the dot product implied in your question, but we can construct a normed vector space from it.