Use of "what" vs "that"

Solution 1:

"That" can introduce an explanatory dependent clause. "What" cannot. "That" indicates a specific quality of the object in question, or a subset of a larger set. "What" indicates the object in question itself, or the entire larger set. (e.g. "What was the bird you used to own that was red?")

In your example, "what" is incorrect because it creates a logic problem; it establishes an equivalency between the two clauses. The second clause, however, is not equivalent to the first; it explains the first clause. "What would you like to do?" is a reference to the entire set of all possible things you would like to do. "That others have told you is impossible" specifies the particular characteristic of the subset the questioner wants to identify.

Notice how I can restructure the sentence so that the two clauses become substantively and logically equal, and THEN the word "what" would be correct, thus: "What you would like to do IS what others have told you is impossible." (I emphasize "is" to point out the equivalency.)