An idea I just had:

Five hundred years in the future the United Earths Alliance has spread out its borders to include several dozen systems and several hundred worlds.  The people are forbidden to travel outside of Alliance territory out of fear of the hostile aliens they have been at war with for hundreds of years, as one group of aliens after another attack us in an attempt to conquer our worlds.  All legal communication within the Alliance is done by quantum entanglement, a method of communication which has virtually no delay and no limit on range.

We are a group of people who gather enough money to by a ship.  We do so for the express purpose of leaving the Alliance and investigating strange signals illegal hyper-radios have detected outside of the Alliances domain.

It would basically be a sandbox, though we as players will know that not all aliens are hostile and that they are trying to contact us.  The Alliance has been invading them to expand its territory sense shortly after humans first entered space.

edit: One thing I forgot to add that I thought would make it even more fun is for us to have Psionic powers which, one the various ships systems are upgraded with a "Psi controller" could be combined with a system to improve it in a way unique to the power.  For example, electrokinesis would make the reactor use the energy it produces more efficient, but telekinesis would make it burn fuel faster and possibly more completely, and pyrokinesis would make it burn hotter, so it could fuse things faster and use different fuels.  Some powers wouldn't effect some systems, like putting someone with telekinesis on communications.  A telepath on communications would increase the range or efficiency of the system, though.