New Yorkers are used to living in shoe box apartments. Now, more of them can see what it is like to live in something even smaller. Leasing begins Monday at Carmel Place, the city’s first micro-unit development, a nine-story, modular building at 335 East 27th Street with 55 studios ranging from 260 to 360 square feet.

The development, previously called My Micro NY, has tapped into a desire common among many singles to live alone. The building includes 14 units designated as affordable, for which some 60,000 people applied, or nearly 4,300 applicants per apartment. The lottery for these units was held earlier this month, and winners will be informed in January. The building is set to open on Feb. 1.

“It shows the need that people feel for affordable, private space in the city,” said Tobias Oriwol, a project developer for Monadnock Development, of the number of applicants. Monadnock is developing the building with the Lower East Side People’s Mutual Housing Association. Most of the affordable apartments will rent for $950 a month to tenants who meet income restrictions, less than half of what will be charged for market-rate apartments.

Apartments in New York City ordinarily can be no smaller than 400 square feet, but the city waived those restrictions for this development. A zoning proposal by the Department of City Planning could open the door for smaller living quarters, if it is approved by the City Council. It calls for eliminating the 400-square-foot minimum to allow for smaller apartments and loosening some density restrictions to fit more units into buildings. But even if these changes prevail, a building consisting entirely of micro units would still be illegal. For the foreseeable future, Carmel Place will remain an outlier and something of a social experiment.

The development is the product of a 2012 design competition intended to address one of the city’s more vexing housing problems: How do you build safe, legal and reasonably priced apartments for single New Yorkers who do not want to double — or triple — up with roommates?

Carmel Place answers that question with studios that were prefabricated in the Brooklyn Navy Yard, delivered by truck over the Manhattan Bridge and assembled on site in Kips Bay. Kitchenettes are outfitted with mini-refrigerators, two-burner electric stovetops and microwaves in lieu of ovens. Bathrooms are large enough to accommodate a wheelchair, but have stall showers instead of bathtubs.

Of the 12 market-rate apartments that will be available for leasing on Monday, eight will be furnished, and all will be on the second through fourth floors. Furnished studios will each include a pull-down bed, sofa, cabinets and tables.

Renters will pay a premium for a furnished unit. For example, a furnished 355-square-foot apartment on the second floor is listed at $2,910, while an unfurnished 360-square foot unit on the same floor is listed for $2,750 — a $160-a-month discount. The lowest-priced unit listed, at $2,540, is a furnished 265-square foot studio on the third floor. The remaining market-rate units, including a 323-square-foot studio on the eighth floor, with a 268-square-foot terrace, will become available over the next few weeks, and priced based on how quickly the first apartments rent, according to a spokeswoman for the developer.

Small apartments are not new to New York. Thousands of apartments that predate the city’s 1987 zoning restrictions would be considered micro units by today’s standards. In Manhattan, some 3,000 apartments measure less than 400 square feet, according to Jonathan J. Miller, the president of the Miller Samuel real estate appraisal firm. Many of them are tucked away in prewar buildings, some converted from hotels or rooming houses.

“There’s all this concern and wondering, ‘Is it going to be accepted?’ ” Mr. Miller said of the new micro units. “But it’s really not about the physical size, it’s about how they’re priced.”

The smallest units at Carmel Place are about half the size of an average studio in Manhattan, which was 550 square feet in October, according to a report by Douglas Elliman. The median rent during the same period was $2,555, about the starting rent at Carmel Place. So renters will be paying considerably more rent per square foot for these micro units. But for some renters it might not matter — their rent check is about the same, even if the size of the space is smaller.

“It’s like buying a Prius, it’s a niche,” Mr. Miller said. “This is one of those things that the market will determine ultimately whether or not they’re accepted.”

Forty percent of households in New York City are not families, according to census data, yet most of the city’s housing stock is designed to house families. Studios account for only 7 percent of the housing stock, according to the Citizens Housing and Planning Council. Single New Yorkers often subdivide apartments, squeezing several roommates into configurations that are often illegal and unsafe. Adding locks to bedroom doors can violate fire codes. And a provision in the city’s housing maintenance code prohibits more than three unrelated people from occupying the same apartment, a rule that is often ignored and considered antiquated.

“There is this idea that bigger is better, and that we need housing for families,” said Sarah Watson, the deputy director of the Citizens Housing and Planning Council and manager of its Making Room initiative, which ultimately led to the city’s micro-unit competition. “But people change, lifestyle changes, technology changes and the housing needs to change.”

The micro units’ design, by nArchitects, tries to resolve the potential for claustrophobia with ceilings that are more than nine feet tall and sliding glass doors that open onto Juliet balconies. The building also provides communal space, including a gym, two lounges and an outdoor courtyard. A lounge in the cellar will have a pool table and a television, and the eighth-floor lounge will open onto a shared roof deck with a barbecue. “People don’t want to be limited by the size of their apartment,” Mr. Oriwol said.

A communal lounge might make tight living quarters more tolerable, but it does not make it any easier to squeeze a bed (and maybe a table and chair, too) into a tiny space. For that, Monadnock enlisted Stage 3 Properties to offer a brand of furnishings and services that it calls Common. The 17 market-rate apartments are furnished by Common with pieces distributed by Resource Furniture. Among them is a sofa designed by the Italian manufacturer Clei that allows a tenant to take off the cushions and pull down a bed from the wall, transforming a living room into a bedroom. A white lacquer desk can be extended into a dining table that seats 10. Tenants who are unsure about how to decorate such a small space can buy an Common Box, an assortment of décor options like throw pillows, rugs and table lamps.

All market-rate tenants receive an Common amenity package that includes weekly housekeeping by Hello Alfred, an app-based personal butler service; Wi-Fi; cable; and access to events, some of which are free. Tenants living in the 14 affordable units would have to pay $163 a month extra for access to these services. An additional eight apartments, furnished and supplied with the Common amenity package, will be set aside for formerly homeless veterans.

“The market has already decided that space is just one attribute that renters consider when they’re looking for housing,” said Christopher Bledsoe, a founding partner of Stage 3 Properties. Other attributes like housekeeping and free Wi-Fi, he said, might convince some renters to pay more for less room.