Whereas I don't know of any recent progress in this problem, let me mention one result for 
*closed* curves.

> **Theorem.** A closed plane curve of length $L$ and curvature bounded by $K$ can be contained inside a circle of radius  $L/4 - (\pi - 2)/2K$. 

This was proved in 1974 by H.H. Johnson ([link 1][1]) who used calculus of variations methods.  A geometric proof was given a bit later by Chakerian, Johnson and Vogt ([link 2][2]).



------------------------------------------------------------------------------------------

**Edit.** Apparently the problem is still open. Here's an article ([arXiv link][3]), which contains a survey of some known results as of 2009. From the Introduction:

> In 1966, Leo Moser  asked for the region of smallest area which can accommodate
every planar arc of length one. The problem is known as “Moser’s worm problem” and is a variation of universal cover problems. In Moser’s problem, a cover is a set which contains a copy of any rectifiable planar arc of unit length, and is usually assumed to be convex. Such a minimal cover is known to have area between 0.2194 and 0.2738. However, the
original problem remains unsolved.



  


  [1]: http://www.ams.org/mathscinet-getitem?mr=0348631
  [2]: http://www.ams.org/journals/proc/1976-057-01/S0002-9939-1976-0402611-2/home.html%20
  [3]: http://arxiv.org/abs/math.MG/0701391