This photo of Pripyat today makes it pretty clear "how" they did it: lots of fairly large (`10 story) apartment buildings, in classic Soviet-style cheap, massed housing. Pripyat was a planned city, created by the Soviets specifically to service the huge Chernobyl reactor complex. They built the first apartment building there ins 1971, just as the plans for the reactor complex were being put together. The apartment model wasn't accidental: they planned the city as a model socialist community, and banned private cottages and separated housing.

They had originally intended Pripyat to only have about 17,000 people, but a variety of factors attracted Soviet citizens to it. For one, there was severe housing shortage in much of the USSR at the time, and Pripyat was a place where you could get new housing immediately. There were also consumer and luxury goods that were available more easily in Pripyat than elsewhere, as part of the lure. It also was designated as a special Komsomol project, which meant that young people were especially incentivized to move there. There were also some nice amenities, like two stadiums and two swimming pools. So it grew much faster than they had expected, particularly with young and unmarried people; the average age of the population at the time of the accident was 26.

So you can imagine that if you were a young Soviet citizen, squeezed into a tiny, decrepit Khruschev-era apartment in Moscow, or toiling in some underpopulated country town with little access to sausages or cheese, or just generally feeling stale, that a move to a new, happening, walkable high-density city full of new job opportunities, new housing, lots of young people, and surrounded by beautiful forest, would sound pretty appealing.

The footprint of Pripyat is about a square mile, give or take a little. So saying it was 50,000 per square mile is not that off the mark. That's pretty high, but it's not as unimaginable as it might sound. It is on par with the population densities of the areas around New York City — other urban areas that have lots of apartment buildings, but not necessarily skyscrapers. The city in which I work (Hoboken, New Jersey) is a mixture of some very high-density (13 story) apartment buildings), some high density (3-5 story) apartment buildings, and a lot of medium density (1-3 story) buildings that are either shared or individually owned. It is certainly quite different from the suburban sprawl of most American towns, especially those in the midwest, but what it really means is that separated housing is rare, yards are small (more parks), and pretty much every possible urban lot is used efficiently. Hoboken is the way it is because of a few hundred years of urban history (and having been settled largely before the automobile, which meant shorter distances were important), but you can imagine how the Soviets would create a planned city of that magnitude if they were starting from scratch. You can also imagine the why: it is cheaper to build things at that scale, it is more efficient from an energy and equipment standpoint, and the goal was to create something of a "company town," for which a sort of common identity would be reinforced by the living arrangements.

My information on the history of Pripyat comes from Serhii Plokhy's Chernobyl, chapter 2.

It is also worth noting that the Pripyat, much like many other satellite cities build in the late 1960s and early 1970s near various industrial complexes, especially the hydroelectric or nuclear power plants, were not especially densely populated. On the contrary, the general idea of urban planners was to create residential districts that would have been built-up sparsely, with a lot of walking space and green areas between separate houses, especially when compared with urban areas dominated by 19th-century townhouses in large cities. This approach was at least partially inspired by the late-19th century concept of a 'garden city' popularized by Ebenezer Howard and its reception by the Soviet Urbanist school of the 1930s.

At a first glance, housing in Pripyat seems to be dominated by the prefabricated concrete slab apartment blocks with 9 residential floors, usually belonging to 111-84 and 111-121 series, with 16-floor towers of II-60, BPS-6 and 121-60-25 series (the latter are these that somewhat resemble a swastika when seen from the top [well visible on the bottom part of the picture attached by u/restricteddata], and were especially popular in such type of cities). On average, each section of such apartment complex (with a separate entry) had 4 apartments per floor, what meant that a single 121-60-25 apartment tower could fit 36 families (say 150 people, as based onthe population to residence number ratio in Pripyat) on a footprint of roughly 250 square meters. For comparison's sake, an average size of a lot in American suburb of 1980's was 650 square meters (what corresponds to 5-6 homes per acre), meaning that the population density in any neighbourhood dominated by similar concrete slab apartment complexes, whether in USSR or in the satellite states, was 100 times higher than in contemporary American suburbs.

Of note is also the fact that the city also included a common housing in the form of resident halls (not dissimilar from the ones on university campus) for the unmarried people (potentially designed to provide housing for over 7000 people) and several family resident halls with 1200 rooms meant as a temporary lodging for newcomers who e.g., hasn't finished the formalities related to the permanent residence or received a flat in a building that hasn't been finished yet. Although in large cities such facilities usually played a marginal role, in new cities they accounted for a substantial percentage of steadily developing housing (if completely full, the residence halls would account for almost 15% of the planned top population of 75.000 people).

And to complement the information about banning the construction of the individual housing, it is worth noting that this was not a characteristic trait of Pripyat or even newly located cities, but rather a result of a statewide legislation. Although citizens were able to buy a lot from the state and build their own house since 1920s, this opportunity was cancelled by the Resolution no. 561 of the Central Committee of CPSU and Council of Ministers on the individual and cooperative residential housing of 1st June 1962.