The crowd gathered at the SK Olympic Handball Stadium, where votes were counted from Wednesday's elections to pick mayors, provincial governors, county officers and members of local assemblies, Reuters witnesses said.
The Yonhap news agency reported unofficial police estimates of the crowd size.
People chanted and held up placards with the slogan "Election re-run!", along with national flags.
Seoul police did not immediately respond to a request for comment.
About a dozen protesters, most in their 20s or 30s, told Reuters they had gathered at the stadium after seeing YouTube videos or social media posts about the vote counting and complaints.
"I was watching the election in real-time, and when I saw the reports about the lack of ballot papers, I thought this cannot stand," said Lee Ung-yeong, a 21-year-old Seoul resident.
The National Election Commission's "explanations are inadequate, so I came after work".
Another protester, 30-year-old Park Gui-nam, said: "This is a violation of the right to vote."
The election commission and the government are yet to comment on the demands for new elections.
Hours earlier, the commission's head said he would step down to take responsibility for the shortage of ballots in some districts.
Roh Tae-ak said there could be no excuse for a failure that harmed the public and the democratic process, understandably causing distrust.
Fifty of 14,300 polling stations ran out of ballots and voting was temporarily suspended at 22 polling stations due to delays in receiving supplies, Yoon Jae-soo, head of the commission's election policy office, later told a briefing.
Because of high turnout in two days of early voting last week, ballots were printed for only half of voters eligible to vote on Wednesday, Yoon said.
He said there were ballots for 73 per cent of eligible voters across all three days of voting and final turnout was 63 per cent.
During voting on Wednesday, voters at some polling stations waited for hours into the night to cast ballots after polls officially closed.
At one station in Seoul's Songpa district, an angry crowd blocked officials from taking ballot boxes to the stadium after voting ended.
At that station, protesters remained in the rain until Friday morning, when hundreds of police escorted commission officials to retrieve the last two ballot boxes before the count officially ended on Friday afternoon.
Roh, a Supreme Court justice who has by convention led the election commission, said outside experts would be asked to investigate, and he would accept the panel's conclusions.
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