The cans in the market suddenly stopped having labels. I don't think it was because they ran out of paper.
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The roads into the Exclusion Zone are guarded by Ukrainian police officers at checkpoints and you need special permission to get in. But, since most of the area is forested, it is still relatively easy to sneak in unmolested.
In the first months after the accident, police officers in respirators and plastic suits were sent in to stop looters while the soldiers and liquidators were doing their work. Their orders were to shoot thieves on sight. Sometimes this happened; but much of the time they went uncaught, and, even worse, some policemen and soldiers would make deals with them: if said thief bribed the guard with enough vodka, he was able to take virtually anything so that he could potentially bring it to a pawn shop in Kiev.
In addition to this, the government lied to everyone in the Ukrainian and Byelorussian SSR's on television, showing military dosimeters measuring food on stands in Pripyat and coming out clean, people tanning on the Pripyat river, and just generally saying that everything was fine. (Military dosimeters were used for the measuring of background radiation, not individual products.) Even worse was the fact that many parts of the country did not know that an accident had occurred at all.
Because of the accident, many of the farms in Ukraine were unusable due to high levels of radioactive contamination, but that didn't stop them from being used anyway. Workers in "dirty" fields got paid twice as much as workers in "clean" fields, and it got to the point that no one wanted to work in the "clean" fields. In addition to this the beef from contaminated farms was not fit for human consumption because much of it had an isotope called cesium-137, so the government mixed this into sausages with beef from other areas and raised the price so that people would eat less of it.
Prior to about the year 2000, it was illegal for anyone but the guards and plant workers to live within the 30-km radius. However, in a steady stream, the elderly returned to their homes and villages; in the words of one, "I would rather die here from radiation than die somewhere far away from homesickness." Today elderly people are permitted to stay, though it is illegal for young adults and children.
Given that most of the Exclusion Zone is forested, it is easy enough for criminals to theoretically sneak in and hide. Though few cases have been reported by policemen who patrol the area, many stolen cars can be found.
Twenty-one years after the accident, a computer game called S.T.A.L.K.E.R.: Shadow of Chernobyl was released to consumers by a Ukrainian company. The stand-alone game quickly became a trilogy, and tourism to the Chernobyl zone increased dramatically. However, not all of it was legal; in recent years, many foolish people ventured into the restricted area dressed as "stalkers" from the game, and military guards arrest many of them in Pripyat.