I was reading something about a week ago and there was a line that said something to the effect of "the space of polynomials is dense in $L^{2}(\mathbb{R})$" and then there was another line that said that $L^{2}(\mathbb{R})$ is the completion of $\mathcal{C}(\mathbb{R})$. I completely understand why both of these are true for $L^{2}([a,b])$ but I feel near certain that this must be wrong for $L^{2}(\mathbb{R})$. For example, take some polynomial say, $x^{2}$ and then $$ \int_{\mathbb{R}} \left(\lvert x^{2}\rvert \right)^{2} \, dx = \infty $$ and some continuous function with infinite support $$ \int_{\mathbb{R}} \lvert \mathbb{1}_{[0,\infty)}(x) \rvert^{2} \, dx=\infty $$
So am I missing something or was this just a careless error on the part of the author? Also, if this is false, how then for $L^{2}(\mathbb{R})$ (or any $L^{p}$ space over $\mathbb{R}$) are these "holes" filled?
You are correct. Polynomials are not square-summable over the real line (or a half-axis), and thus are not (elements of the equivalence classes which are) members of $L^2(\mathbb R)$.