I would rather say "How was IT possible THAT an animal without...." But, having said that, it's correct to have multiple "that" in one sentence. For example: "I thought that that that that that man had written was wrong" is grammatically correct.

Also "was" is past tense so the verb "become" should also be past tense -> "became" -> "How was it possible that an animal without feathers or flight muscles became a bird?" (Or in all present tense -> "How is it possible for an animal without feathers or flight muscles to become a bird?")