= Solution
Choose $\phi\in C_c^\infty(\mathbb R)$ with $\lVert\phi\rVert_2=1$, and for $R>0$ set
$$
u_R(x)=R^{-1/2}\phi(x/R).
$$
A change of variables gives
$$
\lVert u_R\rVert_2=1,
\qquad
\int_\mathbb R|u_R'(x)|^2\,dx
=R^{-2}\int_\mathbb R|\phi'(x)|^2\,dx.
$$
The latter tends to zero as $R\to\infty$. Since the energy is nonnegative,
$$
\boxed{
\inf_{\substack{u\in H^1(\mathbb R)\\\lVert u\rVert_2=1}}
\int_\mathbb R|u'(x)|^2\,dx=0.
}
$$
This is <vanishing Dirichlet energy by dilation on the line>.
The infimum is not attained. If an admissible $u$ had zero energy, then its weak derivative would vanish almost everywhere, so $u$ would be constant almost everywhere. The only constant in $L^2(\mathbb R)$ is zero, contradicting $\lVert u\rVert_2=1$.
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