This is a very nice problem! The answer is quite a surprise: $ \sqrt 2$. To see that $ \ell\geq\sqrt 2$ consider a square $ ABCD$ and translate the triangle $ ABC$ by $ \overrightarrow{v}=\epsilon\overrightarrow{BD}$, obtaining hexagon $ A'BB'CC'D$. Any strip covering this hexagon must be of width close to $ \sqrt 2$, and by varying $ \epsilon$ one can make such width close as we want to $ \sqrt 2$. Now to see that a strip of width $ \sqrt 2$ cover any hexagon $ UVWXYZ$ from $ \cal F$ consider the triangle $ UWY$ obtained by connecting alternating vertices from the hexagon. If one side of $ UWY$, say $ UW$, has length not exceeding $ \sqrt 2$ we are done, since the distance between opposite hexagon sides $ ZU$ and $ WX$ does not exceed $ UW$ and we can cover the hexagon with a strip of width $ UW\leq\sqrt 2$. So suppose that all sides of $ UWY$ are larger than $ \sqrt 2$. Since its smallest altitude must not exceed $ 1$, one can see that $ UWY$ is obtusangle: if the smallest altitude $ YT$ is perpendicular to $ UW$, then $ \cos\angle TYU <{1\over\sqrt 2}\Longrightarrow\angle TYU > 45^{\circ}$ and, analogously, $ \angle TYW > 45^{\circ}$ and, finally, $ \angle UYW > 90^{\circ}$. Let $ r$ and $ s$ lines parallel to $ UY$ and $ UW$, respectively, with $ d(r,UY) = d(s,UW) = 1$ . It's not hard to see (but it's not so easy either!) that $ X$ must belong to the rhombus determined by the four lines $ UY$, $ UW$, $ r$ and $ s$; otherwise, either the hexagon is not convex or one of the triangles $ XUW$ and $ XUW$ have all altitudes larger than $ 1$. Since $ \angle UYW > 90^{\circ}$, $ UX <\sqrt 2$ and the opposite hexagon sides $ UV$ and $ XY$ are less than $ \sqrt 2$ apart and we can cover the hexagon with a strip of width $ \sqrt 2$ again.

The hexagons must be convex hexagons! you can check it in the official web http://www.oei.es/oim/prova2es.pdf You need to consider the hexagon D'A'ABCC' to obtain that \ell\geq\sqrt 2

what is to translate a triangle(ABC)?

this seems very similar to this japanese problem http://www.imomath.com/othercomp/Jap/JapMO07.pdf .... quite curious, huh?