XVII OBM

The answer is $2L$. Consider the following net of the tetrahedron:



Let $P$ be a point of one of the edges of the tetrahedron. The loop will pass through $P$ some time. But $P^{\prime }$ on the net coincides with $P$ on the tetrahedron, so by the triangular inequality the loop must be at least $P{P}^{\prime }=2L$ long.

It is possible to maintain the loop on the net parallel to $A{A}^{\prime }$, so the least value is $2L$.