TL;DR NO in a sense that, you can't do PvP, You can't see other players, You can't explore with your buddies, etc, etc. and YES in a sense of that there is an online function to open your Galactic Map and Scan here and there, and actually see traces of other players, and by traces I mean stuff they have maybe dropped in a planet, or creatures they named or planets
This is coming directly from Sean Murray (Game Director for Hello Games®):
To be super clear - No Man's Sky is not a multiplayer game. Please don't go in looking for that experience.
No There is Not.
For the past few days I have been encountering people wanting to know a definite answer for this, and the definite answer is that there is actually NO such thing as multiplayer, when I first came in here I barely knew about the game and I know I shouldn't had provided an answer where I have limited knowledge to, however my impulse in wanting to help everyone seemed to be the motivation in getting involved in this, and because games are part of my passion as well.
Now I have separated myself in the past few hours just to purely do research based on this subject, And let me just begin with, Even though the Developer mentioned and could almost had swear before the heavens that the game indeed was gonna bring some "form" of Multiplayer capabilities, it was totally being misunderstood and misinterpreted by many people, Now I have came up with the conclusion that, NO, there is no Multiplayer in this game, basically the developer himself said it in many interviews, that if you would go with your friend and maybe gather in some point, the probabilities of you seeing him and vice-versa are extremely narrowed down, next to null.
Because of the massive procedural settings the game brings, Therefore, by the time you both try to accomplish such mission, it could have been ages already, and either you or your friend will drop off from keep on working on such thing.
And even though and surprisingly enough two players have actually done that, Take a look at this article backing up my statement:
...Throughout the stream, viewers have seen the players stand in the same space stations, speak with the same planetary denizens and stop by identical waypoints, only to never actually see each other. Also of note is that one player's planet is experiencing daytime while the other is venturing during the night...
Article by Polygon
Even though there was an update to the article, it still remains that it's not possible that there is multiplayer on the game... Not to mention that you can actually pause the game, everything stops moving, creatures, even ships flying by, doesn't that gives you a sense of a static feeling?
Yes There is.
Also this is where most people are getting confused, because Yes while there are some multiplayer features built-into the game, like with many other Singleplayer games, they are ONLY and strictly dedicated in helping the player to find traces of other players, like for example if they dropped something in a planet, and you later come in, you can see the dropped equipment, and if you go to a planet you will see if it was discovered by someone already, see creatures named by other players etc, etc. and there is no motivation behind the developers whatsoever to change that AFAIK.