No Man's Sky is aiming for. The game is meant to be about exploring a strange and vast universe of procedurally generated planets. In a move that should excite MMO fans, the game will apparently include a shared playing space for everyone.
The game at its heart features a shared space for all players, with the impact of one person's actions having a ripple effect. What form of interactions you'll be able to have remains to be seen, but the developers behind the game want to offer the idea that pure exploration is a dangerous and hostile affair in environments that may not welcome you. As the game progresses through development, we'll be keeping an eye on it, since the idea of a pseudo-MMO allowing you to trek across the endless darkness of space and interact with others is a heady once -- even if your interactions are only through the consequences of the other person's passing.
[We've now added the trailer after the cut.]