All living things have DNA. That’s the set of directions that makes them work. Archaea (formerly Archaebacteria) are alive. So they do have DNA, but like the bacteria, their DNA is not in a nucleus. The DNA is not surrounded with a membrane like it is in the cells of plants, animals, fungi, algae, and protozoa.

All living things either have cells or are cells. The Archaea and bacteria are called prokaryotic because of the lack of a nucleus, their small size, and the lack of any membrane besides the main membrane. Other cells are called eukaryotic. They have nuclei and organelles (important structures) that are covered with membranes. They are also larger.

You might be interested in a career in microbiology.

Yes, archaea have DNA, as do eukaryotes, bacteria, and a new group I am not aware if it has a name yet.

All living things have DNA, and some non-living things like viruses do too.

Archaebacteria in many ways are similar to bacteria. The main differences are that their genes and the way they metabolize food are more closely related to eukaryotes. Every living thing needs a way to propagate their genetic information from mother to daughter cell and DNA is the standard form. DNA is really chemically stable; so much so that we’ve picked DNA off of cavemen. Some viruses have RNA instead of DNA as their genetic material, but RNA is much less stable and is easily destroyed.

By the conventional understanding of life, if you found a system that replicated itself without DNA or RNA, it would not be considered living.