It would help to know what $L$ is.
update This answer was for the question of how short the longest projection could be if $L$ is allowed to vary. There have since been some clarifications.
Here is an upper bound which I think I can prove to be the minimum among all the $d$ dimensional spaces of $\mathbb{R}^n$ (with $n>d$ of course)
Let $L$ be the subspace of $\mathbb{R}^n$ consisting of vectors whose first $d+1$ coordinates sum to $0$ and whose remaining coordinates are $0$. Project the vectors of $\lbrace0,1\rbrace^n $ onto $L.$ The length of the longest projection is
$$\begin{cases}
\sqrt{\frac{d+1}{4}}, & \mbox{if }d=2k+1 \\
\\
\sqrt{\frac{d+1-\frac{1}{d+1}}{4}}, & \mbox{if }d=2k.
\end{cases}$$
