When Rebecca Cooke introduces herself to voters in western Wisconsin, she notes how rough she and her working-class family had it when she was growing up.

“In high school, we had to sell our cows because of the price of milk and competition with other dairies,” Ms. Cooke, who is challenging Representative Derrick Van Orden, a hard-right Republican, said at a recent town hall.

Bob Brooks, a retired Bethlehem firefighter who is running against Representative Ryan Mackenzie, a Republican, in a Pennsylvania district that is one of the most competitive in the country, makes a similar pitch, describing how he was raised by a single mother who worked as a bartender.

“Pizza delivery, dishwasher, bartender, drove a beer truck, landscaper, coached baseball and yeah, I was a firefighter,” is how Mr. Brooks laid out his biography in his campaign launch video.

Even that sounds coddled and comfortable compared to the life story of Nathan Sage, a car mechanic who was raised in a trailer park in Mason City and is now running an outsider’s campaign for Senate in Iowa.

In a populist moment when voters are angry at a government that many believe has failed them at every turn, a slew of working-class Democratic candidates are entering competitive congressional races across the country with an appeal that appears aimed at being relatable, if not particularly uplifting: Our lives are just as difficult and infuriating as yours.