Suppose we have more than one chord. (it's trivial otherwise). Let $S$ be that union of arcs on the circle cut off by the given chords. The total measure of $S$ is less than $2\pi/6$. Let us randomly pick up a regular hexagon $A_1 A_2\dots A_6$ inscribed into the circle. The probability that $A_i\in S$ is less than $1/6$ for any $i=1,2,\dots,6$. So, the probability all $A_i,i=1,2,\dots,6$ are outside $S$ is positive. Thus, there exists a position of the hexagon with required condition.