Where did the cell membrane of Archaea come from?

Structural diversity among archaeans is not limited to the overall shape of the cell. Archaea may have one or more flagella attached to them, or may lack flagella altogether. The flagella are hair-like appendages used for moving around, and are attached directly into the outer membrane of the cell. read more

The OP asks whether cell membranes eventually formed around self replicating molecules as a protective layer. In that sense, the question incorrectly assumes that archaea were the first living entity to form. read more

Archaea and bacteria have generally similar cell structure, but cell composition and organization set the archaea apart. Like bacteria, archaea lack interior membranes and organelles. Like bacteria, the cell membranes of archaea are usually bounded by a cell wall and they swim using one or more flagella. read more