New Yorkers played central roles in developing such technological innovations as air-conditioning, toilet paper and the atomic bomb.

But when confronted with one of civilization’s most ubiquitous inventions — the plastic garbage can — many residents of New York City find themselves positively stumped.

This conundrum became acutely apparent last month, when a new city rule went into effect requiring all residential buildings with nine or fewer units to put their trash out in bins that have secure lids. Similar rules took effect last year for restaurants, some of which have struggled to comply.

Last week, Mahmoud Kasem opened a pair of rusted steel doors in the sidewalk in front of his business, the Al-Aqsa Bakery & Restaurant in Bay Ridge, Brooklyn, and descended into the mucky darkness of a small basement.

By squishing, stacking and cramming, Mr. Kasem can squeeze an almost infinite number of black plastic trash bags into this underground space. But bins — particularly the wheeled ones the city now requires restaurants and smaller apartment buildings to use — can’t be squished or stacked. His bakery generates enough trash to fill 12 bins three times a week. But Mr. Kasem has no place to put them all.

“I’ll be honest: This garbage thing really stresses me out, man,” said Mr. Kasem, 37. “I want someone to come down here and tell me where I’m going to put all these buckets. There’s just no space.”