Is Continuing the story from Warcraft 3 still possible?

Short answer: yes.

Long answer: a decent number of quests will reference events from WC3, though since the Cataclysm expansion included a revamp of low-level zones there's some odd time travel. As a result...

levels 1-60: dealing with the Cataclysm and the return of Deathwing, who is from WC2 so you may not have heard of him. The naga show up in a few places, particularly a little Darkshore. There are also references to the Old Gods, who you see briefly in the Frozen Throne undead campaign. Nothing major from a WC3 perspective, though.

Levels 60-70: You go back through the dark portal to the Orc homeworld. Quests here deal heavily with Illidan and Kael'thas, as well as the Burning Legion in general. There's also some stuff with the Orcs and Thrall's attempts to contact the ones who stayed behind uncorrupted.

Levels 70-80: Arthas is a big threat now. You storm Northrend to take care of him. Quests deal with him and the undead both under him and Sylvanis. Also some more Old God issues.

Levels 80-85: back to the Cataclysm. Hyjal is open and being healed by the druids, and Vash'ir deals with Naga history and plans (but sadly cut short of a full resolution due to development priorities). Other stuff is interesting, but not directly related to WC3.