Radiation is all around us, all the time. In fact, you receive more radiation from eating a banana than living near a nuclear plant for a year. Other normal activities such flying, x rays and CT scans have a significantly higher radiation dose.
In terms of energy, living near a coal plant as compared to a nuclear plant exposes residents to roughly 100x more radiation as a result of the coal ash.
No energy system has a perfect safety record, but nuclear is statistically just as safe as solar panels and wind turbines and far safer than fossil fuels. In fact, nuclear is estimated to have saved over 1.84 million lives worldwide by offsetting fossil fuels.
In the nearly 20,000 years of combined operation there has only been 3 major accidents: Three Mile Island, Fukushima and Chernobyl. Chernobyl was the only accident leading to deaths from radiation. It occurred as a result of management issues and an outdated reactor design.
No nuclear reactor built in the past 40 years has had an accident. Modern nuclear plants include passive safety systems which work independently of human intervention or the need for backup power. Some advanced reactor designs using different coolants promise further safety benefits. For instance, China recently finished a meltdown-proof reactor.
Nuclear plants are built to be secure from any potential threat. In addition to the use of highly trained security guards, nuclear facilities are built from the ground up with security in mind.
For instance, nuclear containment structures are usually over a meter thick composed of steel reinforced concrete built to withstand the impact of an A380. Nuclear plants are also designed to withstand modern threats such as in the cyber space with highly advanced cyber security systems.
Peaceful nuclear is not the same as nuclear weapons. Nuclear power plants typically use uranium enriched to 3-20% compared to nuclear weapons which require uranium enriched to 90%. Moreover, there are strong regulatory frameworks governed by ANSO and the International Atomic Energy Agency to ensure proliferation does not occur. As a member of the non proliferation agreement, Australia would legally is regardless not permitted to produce nuclear weapons.