BOSTON — It is, without a doubt, one of the most compelling stories that Elizabeth Warren tells.

She was just a girl when her father (“my daddy,” she calls him) had a heart attack. The family lost its station wagon and came close to losing its home. That is when her mother dug out “the dress.” “You know the one,” Ms. Warren says of the outfit that her mother saved for weddings, funerals and graduations. Muttering to herself “we will not lose this house, we will not lose this house,” her mother slipped it on, marched to Sears and won a minimum-wage job to keep their family afloat.