China appears to be developing an over-the-horizon (OTH) radar that can spot large ships (like American aircraft carriers) as far as 3,000 kilometers away, and use this information to guide ballistic missiles to the area,. Such radars have long been used to detect ballistic missile launches, and approaching heavy bombers. Some OTH radars have been modified to take advantage of the flat surface of an ocean, to pick up large objects, like ships. Cheaper and more powerful computers enable such OTH radars to more accurately identify ships thousands of kilometers away.
China's principal weapon would be their DF-21 ballistic missile, equipped with a high-explosive warhead and a guidance system that can home in and hit a aircraft carrier at sea. The DF-21 has a range of 1800 kilometers and normally hauls a 300 kiloton nuclear warhead. It's a two stage, 15 ton, solid fuel rocket that could carry a half ton penetrating, high-explosive warhead, along with the special guidance system (a radar and image recognition system).
It is believed that the Chinese have reverse engineered, reinvented or stolen the 1970s technology that went into the U.S. Pershing ballistic missile. This 7.5 ton U.S. Army missile also had an 1,800 kilometers range, and could put its nuclear warhead within 30 meters of its aim point. This was possible because the guidance system had its own radar. This kind of accuracy made the Russians very uncomfortable, as it made their command bunkers vulnerable. The Russians eventually agreed to a lot of nuclear and missile disarmament deals in order to get the Pershings decommissioned in the 1980s.
The Chinese have long been rumored to have a system like this, but there have been no tests. If the Chinese do succeed in creating a "carrier killer" version of the DF-21, the U.S. Navy can modify its Aegis anti-missile system to protect carriers against such attacks. There are also electronic warfare options, to blind the DF-21 radar. Another problem the Chinese will have is getting a general idea of where the target carrier is before they launch the DF-21. This is not impossible, but can be difficult.
China is developing the technologies, and it's only a matter of time before they can actually do it, or at least try to.