If you mean the scenes of the firefighters and others not understanding what the graphite was (or the others not believing it could even be graphite) — yes, that actually occurred. The firefighters were not informed of the potential risks in part because the plant operators were themselves in denial during the first day of it. There were instances of people touching graphite chunks they found outside the plant and getting very high dosages from them. It is important to recognize that these pieces, while very radioactive, would not otherwise indicate that they were dangerous — without the right equipment you cannot tell what is radioactive and what is not, generally speaking. And the firefighters were not at all trained for how to deal with an exploded reactor (but that is not unique to the USSR; I doubt US firefighters are trained for that, either). Even the plant managers refused to believe that the graphite had come from inside the reactor for about a day; it was just too much to accept. This may sound surprising but we know that the power of denial is a strong one.

As for coming into contact with people affected by fallout: that is not such a problem, however, and the show exaggerates the risk of that. After decontamination (removal of clothing and washing the people), even people exposed to high levels of radioactivity are not "contagious" or dangerous to others. Fallout and radiation just don't work that way. This is one of the more misleading distortions of the show.

The show is a mixture of things that happened and things that did not happen, and it is a mixture of things that are scientifically sound and things that are ridiculous. Sifting through both is very difficult even if you are an expert who has read many books on these topics. Just a warning!