Final answer:
An unicellular organism without a nucleus living in acidic environments would be classified in the domain Archaea, which contains prokaryotic organisms known for inhabiting extreme environments, including acidic ones.
Step-by-step explanation:
Based on the information provided, a unicellular organism without a nucleus that thrives in acidic environments would belong to the domain Archaea. These organisms are known as prokaryotes, the informal name for cells without nuclei, which also include those from domain Bacteria. While bacteria and archaea are both prokaryotic, they are different from each other, with Archaea often found in extremely harsh environments such as hot springs, salt flats, and acidic surroundings. This suits the description of an organism that lives in acidic environments. Archaeans are genetically quite distinct from bacteria and have unique properties, such as differences in their cell membranes and ribosomal RNA (rRNA), that justify their placement in a separate domain.