Is the following consistent?

There exists $X \subseteq [0, 1]$, such that $X$ does not have measure zero and for every $Y \subseteq X$, if $Y$ does not have measure zero, then there are $y_1 < y_2 < y_3 < y_4$ in $Y$ such that $y_2 - y_1 = y_4 - y_3$.

Edit: Under CH, there is no such $X$. This follows from a result of Erdos and Kakutani that says that CH is equivalent to the following statement:

The real line $\mathbb{R}$ can be partitioned into countably many rationally independent sets.

The category analogue has a positive answer.

Maybe I should add a comment to Pietro Majer's response:

It is a simple consequence of Lebesgue density theorem that every non null measurable $Y$ must contain equal distances. There are non trivial problems about avoiding patterns in positive measure sets (Erdos similarity problem) but this one is quite different.