Sasha lives in the Soviet Union of the 1950s, when Joseph Stalin ruled with an iron fist.
A Communist dictator, Stalin wanted the USSR to quickly move from a farming-based society to one that was super industrialized and based on collectivization. As you can imagine, this required tons of control, and the only way to maintain this was by brutal means.
Anyone who questioned the system or its methods was imprisoned as an enemy of the people. Under Stalin, over twenty million people were imprisoned, sent into exile, or executed (Author's Note). To exercise total control, the state had a vast system of prisons, police, and surveillance, which of course made just about everyone afraid for their lives at one point or another. Paranoia ran rampant, and trust was tricky to find.
How does The Big Picture translate into day-to-day living in Sasha's time? Well, for starters, he lives in completely cramped conditions. He and his dad share the komunalka, or communal apartment, with over thirty other people. While Sasha and his dad have a fairly nice room all to themselves, there are others (like Stukachov) who squeeze in five or more to a small room. And still others don't even have a room; they sleep in the hallway or in the bathroom.
This type of living arrangement was said to increase feelings of belongingness to the group, but it's also a super effective way of getting people to spy on each other (which is exactly what happens when Stukachov turns in Sasha's dad).
People living in the Soviet Union during this time didn't have a choice of where to live—that was all dictated by the State. And don't think that they could skip out and escape this by taking a nice shopping trip. No way. All of the stores were also owned by the State, and you had to wait in long lines to even buy basic groceries. So, people couldn't just skip on down to Safeway or Whole Foods to buy what they wanted to eat.
Most people (that is, everyone except for the head honcho communists calling the shots) didn't have much choice of what they ate and usually didn't get nearly enough. Remember when Sasha eats the carrot slowly, so the experience won't come to an end (3.2)? That's a big hint of the food scarcity and lack of variety that was the everyday situation in Sasha's world.
His school situation probably seems pretty strange, too. In what we see, there doesn't seem to be a whole lot of learning going on. Typical subjects like math and science seem to be glossed over, but students get double doses of political ideology and conformity to groupthink. Even the playground games reinforce this: Sasha and his friends play "firing squad" instead of innocent games of tag or, you know, an actual, lighthearted snowball fight.
So, Sasha's day-to-day living experience shows us the practical results of communist control and paranoia that comes from The Big Picture.