Well it's technically an "early access" game despite already having an expansion. The game is supposed to enter "finished product" state and get a retail release sometime this spring.
There is no story per se. You just wake up on an island with an alien device implanted in your arm and there are dinosaurs roaming around. The game doesn't tell you what to do or where to go, your objective is just to survive. There's no quests or NPC's around. It's just you, the animals, the elements, and if you're playing on a PVE or PVP server other players.
They do have a story of sorts in the form of explorer notes. Notes left by another individual sometime before you wake up on the island. The notes "will help players get closer to solving the mysteries behind the titular ARK."
The ultimate objective of survival is to level up to max, and build better bases, armor, weapons, and acquire/breed better animals to penetrate further into the island. Obviously there's more end game going on in a PVP server because of the constant threat of being raided by other players and their tribes (even if you're not online).
Right now just playing solo on a locally hosted server to get a feel for the game. It is super easy to get some trophies in this because you can cheat to get them (when playing on a self hosted server you can use admin commands to do basically whatever you want. Did that to hit max level (100) and force tame a few dinosaurs (including the T-rex and the Giga).
Killed that character though and started over with a new character with all the server stuff set to default official server times/rules/etc to actually play the game in preparation for starting out on an official PVP server.
With a tribe, in either PVE or PVP, we could have a lot of fun working together building up a nice base or even our own little city. Recommended. You guys should consider it.
Rust and Far Cry: Primal I think are pretty good comparisons. Rust for the survival/crafting + multiplayer experience and Primal for the overall feel of the game (IMO, Primal is what ARK would be if made by a AAA team, with a story, and no multiplayer, and I guess no dinosaurs).
The base building in ARK is similar to that of Fallout 4. I've gotten far enough into it to learn most of the crafting engrams, but thus far I've been less frustrated with the building in ARK than I was in Fallout.