There are two primary ways that radiation tends to be damaging to humans. One is by direct external exposure. Radiation (high energy particles or photons) hits or penetrates the human body, causing cell damage and in high enough doses illness or sufficient DNA-damage that it can cause enough mutations in cells to increase the risk of cancer. The other way is to have a source of radioactivity become ingested or part of your body, where it can then subject you to dangerous levels of radiation internally over time.

Let's look at some examples of the first kind of exposure. In 1999 at a processing facility at the Tokaimura nuclear power plant in Japan workers inadvertently placed too much uranium into a precipitation tank and caused it to become critical (essentially become a small nuclear reactor). This flooded the immediate environment with high levels of neutron and other radiation, directly exposing the 3 workers surrounding the tank with very high levels of radiation. The two workers who were basically directly in contact with the tank at the time (because they were actively involved in the operation of pouring stuff into it) received lethal doses of radiation. They were hospitalized and died only months later due to massive internal organ damage. The other worker who was a bit farther away received a high dose of radiation and was hospitalized for several months but survived.

Another more relevant example is the "elephant's foot" in the basement of reactor number 4 at the Chernobyl nuclear power plant. This structure was formed from a significant amount of the reactor's melted fuel and other components which flowed out of the reactor after the disaster. Because the elephant's foot contains huge amounts of fission products that are very radioactive it was initially producing enough radioactivity to give anyone who stood next to it a lethal dose within a matter of minutes. Today the level of radioactivity has fallen off due to the decay of short-lived radionuclides (precisely the thing that made it so dangerous to start with) but it is still very dangerous.

When the Chernobyl reactor number 4 exploded it distributed huge quantities of those same fission products across the Ukrainian countryside and, indeed, the world. The main reason why the exclusion zone was evacuated was because this debris is very hard to clean up. What it represents is particulate matter (ash, mostly) from the reactor, some of which contain small fragments of highly radioactive material. As long as these stay outside of your body they aren't super dangerous to a human, the real danger is if they end up in the lungs or digestive system. They could get lodged in the lungs for an extended period, doing extensive cellular damage up close, increasing the risk of cancer. Or being ingested some of the radioisotopes might become integrated into the body metabolically, such as strontium-90 (which can be taken up as a replacement for calcium in bones) or iodine-131 (which can replace ordinary iodine in the thyroid gland), and then those isotopes will irradiate material around them over your lifespan, increasing the risk of cancer.

This is why there is a danger of living in the exclusion zone. If you were always in, say, a spacesuit you could live in the exclusion zone pretty safely. The average amount of external exposure to radiation is pretty small. And, indeed, many people visit the exclusion zone. The main risk is "winning the lottery" by ingesting or inhaling some little speck of radioactive dust which then increases your risk of cancer. If you were living your everyday life in the exclusion zone it's likely that your exposure to radiation would be much higher than average due to this, and thus your risk of developing cancer or other illnesses would go up.

In contrast, working even in the next reactor over from No. 4 is not necessarily the same level of risk as living in the exclusion zone. If you breathe filtered air, use protective personal equipment (PPE), shower, etc. it can be relatively safe. It's mostly about using the correct PPE and being mindful of the risks. Granted, it is pretty likely that the workers in the other Chernobyl reactors probably were probably put at higher risk due to radiation exposure than the typical workers in a reactor but that was a tradeoff the operators and the government were willing to make, at least for a while.

Ironically since 2000 there has likely been more activity in the reactor complex than when the reactors were in operation. The construction of the New Safe Confinement structure as well as ongoing activities in cleaning up the site, including the No. 4 reactor and the hastily built "sarcophagus" containment building has brought thousands of workers to the area, and today the area around the reactor complex is regularly visited by workers every day. However, these workers are all specially trained, use appropriate PPE, and are heavily and carefully monitored for radiation exposure. Their radiation exposure is probably no greater than that of a typical reactor worker.

Contrary to the common opinion, the Chernobyl Exlusion Zone is generally very safe, usually safer than many industrial areas, especially the various types of dumping grounds. The problem is that within the area of over 700 square kilometers the external part of Zone covers, there are few 'hot spots', where the radiation (usually short-range beta) is quite noticeable and the ground is irradiated and thus radioactive dust particles can be inhaled, ingested or carried outside the zone on the shoes or clothing. These spots are small, few, marked and known to the personnel allowed inside the zone, but if the area was open to the public or settled, the number of people coming in contact with the irradiated areas even incidentally could have became a problem. In other words, the Exclusion Zone is nowhere as dangerous as the tall tales go, but it is not fit for settlement, either. The city of Pripyat itself could have been even technically resettled, as the average background radiation levels are usually in the ballpark of 1.5 mSv/h with the 'hot spots' reaching 30-40 mSv/h. Of course, even the former value is 10 times higher than the background radiation in most places, but the annual buildup would be similar to the one that is usually allowed for the nuclear facility workers. In addition, there are inhabited places that show even higher background radiation, such as Ramsar in northern Iran, where that average value is similar to that of accidental 'hot spots' and ten times higher than general value for Pripyat, with inhabitants showing much lower rate of genetic anomalies and no difference in pathological conditions or life span in comparison with other inhabitants of the region, what is generally considered a good argument for the radiation hormesis theory, but I digress.

Also, as u/rocketsocks already explained, radiation is a serious danger, but it is relatively short-ranged and quite easy to monitor. This means that when the irradiated area has been cut off by the containment facility (initially the Shelter no. 4, commonly known as 'The Sarcophagus' and finished in late 1986, now the NBK or 'The Arch', installed in 2019) and the territory of the plant underwent meticulous decontamination, the radiation within the plant today is roughly the same as before the incident, especially in the facilities located at a certain distance from the reactors.

Now, personnel of the nuclear power plant is well trained in variety of protective and preventive procedures, have good knowledge about the radiation and its potential influence on living organisms or human health and thus are perfectly able to keep away from any dangerous area. Furthermore, the chances of them exposing themselves to any dangerous source of radiation is basically non-existent, as they generally do not leaving the territory of the plant that has been monitored on a constant basis for the last 34 years (save for daily travel by train). Furthermore, the work procedures are usually well-organized in most places like this, and with the Chernobyl power plant being a place of the only INES-7 incident until 2011 (the second being the accident in Fukushima), ad thus the safety measures were very strict. Although resettlement was initially considered to be only temporary, but between the contamination with long-lasting caesium-137 and iodine-131 that was linked to thyroid disorders, effort required to check the actual levels of radiation throughout the area and sheer psychological impact, authorities decided to abandon Pripyat permanently. New city of Slavutych has been erected 48 km (30 miles) to the east and majority of the workers were eventually resettled there. Today, they are transported to the power plant by the special railway linking Slavutych with the plant, without any other stops (partially due the fact that it traverses the territory of Belarus) and essentially arrives within the perimeter of the plant, and thus workers' exposure to radiation is minimal if any.

Outside the 'hot spots', radiation in the zone is generally relatively low and does not significantly exceed background levels typical for Central Europe (that are usually lower than these in Scandinavia, for example) and are in the ballpark of 1.5-2 µSv/h. With such level of radiation, the equivalent dose of 0.15 Sv that is usually considered an upper safe limit for an average human, one would need to remain in this area for 9 years, assuming the natural recovery mechanism were for some reason not functioning (and this alone would possibly kill a them even faster). The area adjacent to the walls of the 'sarcophagus' show the levels of 5-6 µSv/h, a noticeable amount, but still not unusual for various parts of a nuclear facility. Hot spots show noticeably higher beta and gamma radiation, sometimes over 1.5 mSv/h, but this still means that a person would need to stand there for 100 hours straight to cross the safety level of exposure. And even the latter are still relatively freely available for tourists. Really dangerous areas, where entry is explicitly forbidden, such as some sections of Red Forest, the radiation level can reach 10-12 mSv/h. This could lead to an unhealthy exposure after several hours, what is very much possible if one e.g. contaminates clothing with radioactive soil. But such spots are small, rare, and secluded (not to mention that they are marked and most are cordoned off), so the workers of the plant are never found anywhere close to them.

And there is one important thing to be mentioned here. The Exclusion Zone is far from being tightly controlled, depopulated place. Although it has been initially completely evacuated, some people started to return to their homes, primarily in the villages surrounding Pripyat and Chernobyl, as the urban inhabitants were resettled and usually returned to their work at the plant, not to mention that the Pripyat existed roughly 16 years, so people there had no connection to the place, but villagers who lived there for any generations did not share this outlook. Throughout the last three decades, almost 3000 people returned to the zone, but most left this place sooner or later, as living conditions were extreme with no infrastructure to speak of. On the other hand, roughly 80-100 people still live there. Although officially this is against the law, police and administration leave these people be, knowing that they are usually sedentary, elderly people, unlikely to cause any trouble.

Please note that although the last reactor block has been turned off in 2000 (two others were closed in 1991 and 1996 respectively), there are still workers on site, usually technical staff tasked with the monitoring of the reactors to ensure that they do not pose any danger, teams dismantling and disposing of the nuclear devices and facilities, security protecting the plant from trespassers and occasional scientists and engineers using the plant as an unique place for experiments or training. What is interesting, the last block that was decommissioned in 2000 was not only the one closest to the Block 4, where the meltdown occurred, but also shared the building with the latter (Blocks 1 and 2 were located in separate structures, roughly 100 meters away).

In general, people working at the power plant are under strict radiological control, going through radio check-up every day, full medical examination every few months and being eligible to three-week state-funded vacations in sanatorium every year. In addition they enjoy significantly longer paid vacation time, equal to 56 working days in a year, that is extended to 72 days for 'liquidators' i.e. workers who took part in the clean-up after the 1986 incident. This means that they essentially work only slightly over 2/3rd of the year (usually on 8-hour shifts, which means each chance of exposure is interspersed with 16 hours 'downtime') what helps the recovery even in case of potentially unhealthy levels of chronic radiation they might be exposed to. With the working year roughly equal to 1500 hours, it means that these people could have been able to work not only in the plant, but also in many less irradiated 'hot spots' with little to no ill effects, at least from radiation alone.

So, to sum it up, the radiation in the area, although substantial, is generally manageable and the plant itself has been decontaminated and secured to the best ability of the people involved. This, along with the fact that workers generally have no contact with the Exclusion Zone itself and travel to the plant from few dozen miles away.