"We are sleeping for six days."

Translation: 우리는 엿새 동안 자요.

[Phalanx8710]

Is it useful to learn the native Korean counting words past 3 (사흘)? Koreans have told me that only old people ever use them (one friend didn't even know 나흘 was four). Here wouldn't 육일 be much more useful?

[Ash-Fred]

일 is a bound noun, so it is 육 일. While you're somewhat right, they are still used not too rarely. I think a few young Koreans confuse 사흘 with 사 일 because it's both 사, you know, and then they are like "If 사흘 is four days, what is 나흘?" Not to offend anyone, but nevertheless you'd look very uneducated if you didn't know those words (as a native speaker).