I would say the first and last sound the most natural. The other two sound a bit off.

All are understandable, but not equally natural. “I wasn’t even born back then.” Fully natural and the most common. “I wasn’t even born yet back then.” Redundant. “yet” plus “back then” overlap. Not natural. “I wasn’t even born yet at that time.” Grammatically fine, but a bit formal. Still natural. “I hadn’t been born back then.” Correct, but less common. Sounds more formal or narrative. Best choice in everyday speech: “I wasn’t even born back then.”