Let 
$$I_k:=\int_{|x-k\pi|<1/k}x^4\exp(-2x^8\sin^2 x)\,dx.$$
Then, as $k\to\infty$,
$$I_k\asymp k^4\int_{|x-k\pi|<1/k}\exp\{-(2+o(1))(k\pi)^8\sin^2 x\}\,dx \\
=k^4\int_{|x|<1/k}\exp\{-(2+o(1))(k\pi)^8\sin^2 x\}\,dx \\
=k^4\int_{|x|<1/k}\exp\{-(2+o(1))(k\pi)^8 x^2\}\,dx \asymp1,$$
whence 
$$\int_{-\infty}^\infty f(x)^2\,dx\ge\sum_{k=1}^\infty I_k=\infty.$$
So, $f$ is not square-integrable. 

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Reasoning similarly (but using, say, $1/k^{b/3}$ instead of $1/k$ in $|x-k\pi|<1/k$), one can see that for any real $a,b>0$, letting 
$$f(x):=|x|^a\exp(-|x|^b\sin^2x),$$
we have the following: 

 1. $f$ is continuous, but unbounded at $\infty$. 
    
 2.  $f$ is square-integrable iff $2a-b/2<-1$.

In your example, we have $a=2$ and $b=8$, so that $2a-b/2=0\not<-1$, and so, your $f$ is not square-integrable.