Yes, he's literally asking about laundry.
Here's a more complete segment of the script, for reference:
The lobby is empty -save the Manager- who glances up from his
computer... to find a wounded -and quite bloody- John walking towards
MANAGER: (without blinking) Good evenin’, sir.
JOHN: Evenin’. Is the doctor in?
MANAGER: Yes, sir. Twenty-four/seven.
JOHN: Send him up, please.
MANAGER: Yes, sir. Anything else, sir?
JOHN: Depends. How good’s your laundry?
MANAGER: The best, sir, however, I’m sorry to say that... no one’s that good.
John chuckles, sliding a gold coin across to the Manager.
JOHN: No, I thought not.
So, in this scene John is injured and asking for the hotel's live-in doctor (a literal request, not a code - to patch up his wounds.) This is a service that I expect many professional assassins in this world would need, so it makes sense that the Continental would offer one.
The manager asks if he needs anything else. He doesn't really, but since the manager asked, he inquires about the laundry to deal with his bloody clothing. He's not too surprised at the answer, of course. He was asking partly - if not entirely - in jest.