No, it's not impossible.
It's an eternal parenting dilemma, getting kids to eat vegetables and enjoy them. That last part is key because so many of the vegetables served to kids these days are hidden. They come as tater tots or pureed fruit-and-veg pouches, meaning that the kids don't ever experience the vegetables in their intact form, and thus miss out on the flavor (and much of the nutrition). They aren't learning to eat vegetables, but rather fruit sugars and deep-frying oil.
How do you get kids on board with real vegetables? An article in the New York Times consulted several paediatricians for their advice on feeding. The consensus included the following tips:
1. Eat together.
When families dine at the same table, it creates a positive atmosphere that has been shown to result in better physical and mental health down the road. Regarding vegetables specifically, it's a chance for parents to model healthy consumption and enjoyment. A kid can still be given choices, like cucumber slices vs. celery sticks, but should be expected to eat what's served. Dr. Nimali Fernando said,
"When families provide an alternative meal for a child, then he’s learning, 'If I don’t eat that, then I’ll eat something tastier.' But if we don’t cater to that, they will end up eating the food that we put in front of them."
2. Don't spoil their appetites.
Kids fill up quickly, which means that if you're handing over snacks or giving them free rein of a snack cabinet in the after-school hours, there's a good chance they'll have no room left for the real meal. When a kid is truly hungry, they'll be much more eager to eat whatever's put in front of them. Better yet is to strive for an early dinner, to eliminate that time of prolonged hunger, which is miserable for parents and kids alike.
3. Familiarize kids with raw ingredients.
Get them involved in handling vegetables, whether it's growing them, picking them out at the grocery store, helping prepare them for a meal, and serving them at the table. You should even let your children play with them:
"[Let children] use squash cubes as building blocks; slicing a beet open and letting kids 'tattoo' themselves with it or use asparagus spears as paint brushes (use juice as paint or just pretend to paint with water). Wash the food when they finish and you can still cook with it so you’re not wasting good food."
I'd add that consistency is key. Start young and stick with it, even if it takes years. My children eat everything, but they still complain when rapini, mushrooms, and Swiss chard appear on the table. Nevertheless, they know the drill: they must eat at least some of it. Learning to handle less-than-delightful foods graciously is an important life skill, especially for anyone who hopes to travel anywhere in life.