Birnbaum was born in the town of Skola, then part of Austria-Hungary. He apprenticed with his father, a tailor, and emigrated with his family to the lower east side in 1913. Later, they moved to Brooklyn and Aaron enrolled in design school. He opened a women's apparel store then later expanded to design and manufacturing specializing in women's sportswear and dresses.
After retiring in 1957 and feeling a bit at a loss, he expanded his knowledge of sketching to teach himself to paint. He once said, "A picture tries to beautify. So you have to put on a little pepper and salt for people to like it." His works are considered 'memory paintings' in that he does not paint from images but rather from remembering scenes from his youth, recording scenes from memory. He and his work were honored by the Museum of American Folk Art in 1995 to celebrate his 100th birthday.