Let's start by definining what Food Irradiation is. As the FDA defines it: "Food irradiation is a process that uses radiation to control pests (e.g., microbes and insects) in food and prevent spoilage...During irradiation, gamma rays, x-rays, or high-energy electrons pass through the food, destroying or inactivating bacteria and viruses that cause foodborne illness."
The FDA considers food irradiation a safe tool to preserve food, increasing shelf life of food. It is also reported that there are no nutritional changes to the food after irradiating them. Independent studies show that food irradiation may actually chemically change the food (which makes sense to me - how are you extending the life span if you're not somehow changing the food), and that has untold impact on the human body.
As the daughter of a woman who has beat breast cancer THREE times, I try to limit my exposure to radiation. I do the pat down search when I fly, I don't use a microwave, etc. Knowing that my food has basically gone through an X-ray machine is concerning to me. Eating local, fresh, organic is always the best option, whenever available. If you can't access those, then boosting your vitamins and minerals that are antioxidants is a good way to offset any perceived affects of radiation. Why not simply protect yourself? Aren't you worth it?
The Environmental Workers Group released a great piece on food irradiation, https://www.ewg.org/news-insights/news/food-irradiation-grosser-you-think, which breaks it down even more.