Asael Woodruff Price was the last of 12 children born to Isaac and Ann Price. He was born in Round Valley, Rich, Utah on 23 December 1904. He was named Asael after Asael Smith (who was the grandfather of the Prophet Joseph Smith) and Woodruff after LDS Church president Wilford Woodruff.

Asael's father died unexpectedly in 1912, but the family was able to keep their ranch in Round Valley. In 1972, Asael wrote about his life: "As long as I can remember, I wanted to become a teacher when I grew up. I loved books and knowledge and information, and sought after those things from the time I went into school. In the 5th grade until the second year high school, I went to live with my brother, Frank who lived in Laketown. He was very good to me and I was glad I was there because in the spring of the year he was shearing sheep he got typhoid fever and was down for several weeks with phlebitis in his legs and on crutches and unable to perform. I was able to take care of his cows and his irrigation and business things that needed to be done so I was glad I was there. ...I have always been impressed with my ancestry and I am glad that my parents were members of the [LDS] church. My mother [Ann Reed Price] taught me everything in the way of righteousness and I shall be forever grateful for it. She was older than most mothers as she was 46 years old when I was born. So she was older than most mothers are [who have young children] and I shouldered the responsibility of taking care of her in her declining years since I was the last of a large family. She always told me that if I would care for her and be with her until I was 25 that that was all that she would ask. So that is what I did. In summers we would spend them at the old homestead in Round Valley on Bear Lake and in winters we would go to Box Elder School District and spend the winters there. We lived in Brigham City, Corinne, Bothwell, Powell, Deweyville, and Fielding."

Asael married Lola Tufts 4 June 1930 in the Salt Lake Temple and had 3 sons. They lived in Ogden. In his 1972 autobiography, he wrote: "Each day brings a grand and glorious experience for me. I love life, I want to live. And the members of the church, and non-members alike, treat me with assurance, respect and dignity. Each day is precious to me. I look forward to living until I am 100 years old. It doesn't seem to far distant as it used to. I was 76 last birthday. I would like to endure to the end and live righteously."