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The first barrier is cost. Healthy food tends to cost a lot more compared to cheap fast food or junk food. This explains why fast food places vastly outnumber any other kind of eatery.
It's not only the cost, but also the access as well. What I mean by that is that stores selling healthy food tend to be in more well off neighborhoods with more money, leaving poorer neighborhoods with less choice. In a sense, cost is tied to this because it costs more gas money to travel to these further grocery stores.
Another barrier is personal preference. Someone who refuses to eat healthy simply won't eat healthy. To help change their mind, it is ideal to find ways to show that there are delicious healthy food options (not everything has to be spinach and kale for instance).
Also, another barrier is lack of education in terms of nutrition. There are a lot of health concepts that many people don't understand. Food science is definitely a tricky field. Though I think if more people understood the basics then they would be able to know how to eat healthy.
Finally, the last barrier is time. Healthy food tends to need more time to prepare the food in question. Compare this to fast food which is available right away and eaten quickly (often on the go). With people living busier lives, taking on more responsibility/work, there is less and less time to sit down for a healthy meal. A way to combat this is to schedule out some kind of meal plan. Exercise would be included somehow as well.