The two candidates who will fight it out to win election as South Korea’s next president overcame great odds to get where they are. Lee Jae-myung was a teenage sweatshop worker whose family survived on rotten fruits. Kim Moon-soo was imprisoned and tortured for anti-government ​​activism. Both survived weeks of political and legal turbulence that threatened to upend their presidential bid.

Now, as the official campaign for the June 3 poll kicks off on Monday, Mr. Lee and Mr. Kim have emerged as two main contenders. They represent opposite sides of a political divide that is unlikely to be bridged, even though both have promised to pursue national unity if elected.

The election follows the removal last month of former President Yoon Suk Yeol, who was impeached for his short-lived attempt to place South Korea under martial law. As such, the campaign is being fought less over policies and more as a referendum on Mr. Yoon and his right-wing People Power Party.

The party has not cut ties with Mr. Yoon, who is facing trial on insurrection charges. Instead, it has ​veered further to the right by choosing Mr. Kim, Mr. Yoon’s former ​labor minister, as its presidential candidate. When Mr. Yoon’s cabinet members were asked during a parliamentary session in December to apologize for the imposition of martial law, Mr. Kim was the only one who refused to stand and bow.

His main rival Mr. Lee​, 60, has led in pre-election surveys. After winning ​his Democratic Party’s presidential nomination with an unprecedented 89.77 percent of the votes, he said: “I am ordered to end the old era of insurrection and regression and open a new era of hope.”

Both Mr. Lee​ and Mr. Kim​, 73, had to clear last-minute hurdles to run for president, adding to the uncertainty that has pervaded South Korean politics in recent months.​