It’s up to each individual country. The EU isn’t a country. You can usually find the different requirements on the country’s government website. Most European countries do have language requirements to get citizenship, and the other steps can be pretty hard. I have an American friend who moved to Denmark, and she’s had to take intensive language classes, maintain employment for several years, and other steps. And she’s still not close.

You could get special visas that allow you to stay there for long periods, but again you have to look at each individual country.