Some creativity:
Put a disk like object into space that orbits the Sun with the same period as Earth and always shows one face towards the Sun and the other face towards outer space. The temperature of outer space is that of the cosmic microwave radiation: 2.7 K ( http://en.wikipedia.org/wiki/Outer_space ). Use this side to store information. This can be done two orders of magnitude more efficiently than on Earth because Landauer's limit depends on the temperature. Use the other side towards then Sun to collect the energy needed for storage.
Filter out illegal positions and account for basic symmetries. At least 3 orders of magnitude.
You are already have improved 5 orders of magnitude.
If on Earth it takes a year to store bits on the order of the upper bound of the number of legal chess positions, then if you improve upon this by 5 orders of magnitude and store them on a space device described above, the radius of the device should be ( http://www.wolframalpha.com/input/?i=sqrt%28%28area+of+Earth%29%2F%282*10%5E5*pi%29%29 ):
28 km
It can be organized that the device is not too far away from Earth all year, so the speed of communication with the device is not limited by the light barrier in any significant way.
Is it not possible to built a 28 km radius space device? It is hard, but is it impossible?
It is certain that chess can be solved in the weak sense practically and it is very likely that this will happen.
However the weak solution is not of too much interest ( it is almost certain that the starting position is a draw ). Only a strong solution can answer interesting questions about the merits of certain openings and endgames.
If it turns out for example that the king's gambit accepted is a win for black: now, this would be a non trivial result.
I'm also interested in questions like whether white can make a move on the first move which loses ( does 1. f3 lose by force for example ).