The Eagle has landed.

Yeah, it does sound kind of weird. When the NASA mission Apollo 11 landed on the surface of the moon, the landing vehicle was named Eagle, and the boys on board were telling mission control in Houston that they were safely on the surface of the moon. While using the expression would not be incorrect, i think it is fair to say that that is the context in which it is usually thought of. Other native speakers of English might not know what the hell you're taling about, and it might sound odd even to speakers of English who are aware of the context. It mostly would resonate with Americans who were adult, or almost adults, in July, 1969. Even to them, it might sound pretentious.

uh....Houston, we have a problem.

Yup . . . "the Eagle has landed" was only a step in the process. After Armstrong got out of the lunar landing module and flubbed his lines, and than hopped around the surface of the moon, they had to blast off to rendezvous with the command module in lunar orbit. Then they had to blast out of lunar orbit and make it back to dear old Earth. From there, they had to all three crawl into the capsule, jettison the rest of the space craft, line up so that the heat shield would protect them during re-entry. Then they had to drop into the atmosphere at the right angle, at the right spot. It wasn't success until the helicopter picked them and delivered them to the carrier. When all three of them were standing on the deck of the carrier, alive and well, that was success. Apollo thirteen is a good example of knowing what success really is.

"The Eagle has landed" is also an idiom to mean "things are successful/complete"

I see. So it wouldn't be such a good idea to use it in daily conversations.
Thanks.

I have got a feeling that, before the NASA mission referred to, this was a catchphrase in a wartime movie, a coded message sent back to base to say that an operation had been successful.

But I may be wrong about that.

I think Chai's point was that something you think is out of date is still often referred to. We still for instance still say " blow off some steam" even though almost no one alive has ever seen a steam engine.

That may well be true--however, i don't think that one would be advised to use it as the OP suggests.

In the the early 1950's, my father was stati0ned in Alaska. It was not at all unusual for the military, especially after a new assignment, to mess up your paycheck. One day, my father came home from work and said to my mother "the eagle has landed." This meant that his paycheck had finally come in. That was started at least in WWII.

Hello, Holocombe.
What do you mean by 'mess up your paycheck'?

'Mess up' is, my dictionary says, mostly, screw up,
which I don't think you intended to say here.

Would you please help me, a S. Korean, figure it out?
Thanks.

It could mean the process was not working properly. The clerks were moving all the paperwork trying to get the right paycheck to the right person. We were transfering from Buffalo, NY to Fort Richmond, Alaska. Dad missed two paychecks. Finally, when he came home with a long over due pay check, he said to my mother "The Eagle Has Landed." She understood exactly what he ment.

To "mess something up" is to get it wrong, usually badly wrong.

It is a colloquial American phrase not much used here, though widely understood.