Why are NoSQL databases not ACID compliant?
How can a database work well if there is no atomicity, consistency, isolation and durability?


They can't. This is why those features were created.

Sometimes you can work around the lack of these features because you need to scale horizontally or scaling becomes your first and primary concern. You may not need need consistency now, you may be able to handle it later.. You may not need all the data returned in order: you may want the queries executed in parallel over clusters. Or your data may itself be fairly un-valuable.

Look at major products that implement these types of systems though -- generally, they suck and they have lots of errors. Ever post something to your wall on Facebook just to see it disappear and reappear multiple times? Or to have a conversation you're commenting on vanish for an extended period of time and come back? That's hundreds of people working to manage data using "NoSQL" and compiling PHP to C++. It doesn't especially work well. It works, and for most companies that's further than they can get developing an alternative.