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About 27:40 minutes into the movie Kursk, Mikhail is calling his mate Anton in another compartment (compartment 5). Anton said that the reactor's melting down, and the water level is rising fast. Anton also said that they cannot leave the compartment or else it's Chernobyl.

As Chernobyl was stated it must have to be radioactive substances. I know that radioactive decays to its half life by the formula of Ernest Rutherford t1/2, and it needs constant cooling. However the Kursk had Granit missiles. Are Granit missiles also a type of nuclear weapon?

And during the situation where I must say everything was in a messy situation, why couldn't Anton just leave his station? All the crew of the ship couldn't survive at last so why he couldn't he join the rest of the team? What could really possibly have happened if Anton had left his station?

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The K-141 Kursk was a "nuclear-powered submarine" ("2 OK-650b nuclear reactors"). That's probably why Anton is comparing it to Chernobyl. They needed to make sure the nuclear reactors weren't damaged "or it's Chernobyl", giving their lives to safe others.

From the wikipedia article on the disaster:

It collapsed the bulkheads between the first three compartments and all the decks, tore a large hole in the hull, destroyed compartments four and five, and killed everyone still alive forward of the nuclear reactor in the fifth compartment. The nuclear reactors shut down safely.

Shutdown of nuclear reactors

The fifth compartment that contained the boat's two nuclear reactors was built to withstand larger forces than other interior bulkheads. Like the exterior hull, these bulkheads were designed to withstand pressure up to depths of 1,000 metres (3,300 ft). The reactors were additionally encased in 13 centimetres (5.1 in) of steel and resiliently mounted to absorb shocks in excess of 50g. The bulkheads of the fifth compartment withstood both explosions, allowing the two reactors to shut down automatically and prevent a nuclear meltdown and widespread contamination of the sea.

Especially the last sentence answers the question why they acted this way and what they had to prevent - a nuclear meltdown in the Barents Sea.

As a comparison, here is more from the wikipedia article on the the USS Thresher (SSN-593), a US nuclear-powered submarine that sank in 1963:

The U.S. Navy has periodically monitored the environmental conditions of the site since the sinking and has reported the results in an annual public report on environmental monitoring for U.S. Naval nuclear-powered craft. These reports provide specifics on the environmental sampling of sediment, water, and marine life which was done to ascertain whether Thresher's nuclear reactor has had a significant effect on the deep ocean environment. The reports also explain the methodology for conducting deep-sea monitoring from both surface vessels and submersibles. The monitoring data confirm that there has been no significant effect on the environment. Nuclear fuel in the submarine remains intact.

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