one half of the station did not know what the other half was doing.

不能省略what.
省略的前提是：被省略的成分不做句中的成分,省略之后不会产生歧义
那么我们知道what 在例句中是作的doing的宾语,省略了就少了成分.
也不能用that指代,what=sth that ,显然用that的话少了sth,
改成：know the things that the other half was doing.就可以了
希望能帮到您what不是作know的宾语吗?而what后面的句子就修饰what.您好，这个复合句可以分为：主句：one half of the station did not know从句（宾语从句）：what the other half was doing.从句作know的宾语，不单单是what。在宾语从句中，what作doing的宾语，就像what are you doing？中 的what一样。希望能帮到您