Consider any functional $f$ which does not attain its norm on
$L_1[0,1]$ (such $f$ exists by James's theorem, but in this case
one can find it without, as an $L_\infty$-function with essential
supremum equal to $1$, which is not attained on a set of non-zero
measure) and let $Y$ be the kernel of $f$. Let $x\notin Y$, if
$\hbox{dist}(x,Y)$ is attained at $y\in Y$, then $f$ would attain
its norm on $(x-y)/||x-y||$.