Please watch the Chernobyl series that was released recently and take note of how easily the radiation escaped the plant and the havoc it created. The suits of first responders were so radioactive they were put in a basement, set ablaze, then sealed up - only to still hold radiation today in 2019. The radioactive ash that flew from the plant killed everything downwind of the flames, rendering land uninhabitable and waters poisonous. All the water used to try and put out the flames became toxic enough that just dipping a toe meant lethal cancer. The vehicles involved in saving people - all blasted with radiation and left unusable. Indeed, a fair portion of all steel in the world is so radioactive from background exposure that it can't be used in construction.

Just tackling the issue of a failing nuclear plant means sacrificing so much, if one were to reach such a state and not be contained, it would very easily consume much of the world over. Clouds of contaminated water could rain down radiation, bypassing borders because you can't police the clouds. One of the proposed solutions was to encapsulate the core and throw it in the sea to cool it to manageable levels, ignoring the fact that would contaminate all of the USSR's drinking water.

In short, yes, a power plant gone awry can poison the planet.