Neither is necessary, but the sentence is best altered a bit to make its meaning more clear. The structure of the sentence is: "The man's real dream is" NOUN#1, NOUN#2. The two noun clauses are NOUN#1 = "achieving..." ("That of" plays no essential role.) NOUN#2 = "rising to a loftier place..." ("Of" plays no essential role.) The reason you can eliminate these words is that "achieving" and "rising" are gerunds that are easily capable of leading their own noun clauses. However, the sentence is a little bit awkward. You need to better link your two noun clauses to the rest of the sentence. One way to fix it is to replace "that of" with "both" and replace "of" with "and": "The man's real dream, as critics suggested, is both achieving a new status and rising to a loftier place in the mysterious hierarchy of human worth." I also eliminated a few words so there wouldn't be too many "and's". I also think the sentence would be stronger if you used infinitives instead of gerunds: "to achieve" and "to rise".

It's written in a very formal register. I can think of a couple of ways that you could easily tone down the formality with minor changes: The man's real dream, as critics suggested, is achieving a new status and a new essence, and rising to a loftier place in the mysterious hierarchy of human worth. The man's real dream, as critics suggested, is to achieve a new status and a new essence, and rise to a loftier place in the mysterious hierarchy of human worth.

Yes, both "that of" and "of" are necessary in the given sentence. "That of achieving a new status and a new essence" is a noun phrase that serves as the subject complement of the sentence, which means it renames or describes the subject "the man's real dream." Similarly, "of rising to a loftier place in the mysterious hierarchy of human worth" is a prepositional phrase that acts as an adjectival modifier of the noun phrase "that of achieving a new status and a new essence." Therefore, both "that of" and "of" are necessary to convey the intended meaning of the sentence.