Archer went to the Klingon Homeworld, though even if they are ignoring that canon, it never seemed very far away - Kirk went quickly there in The Undiscovered Country, and there are published, authorised maps that have it close to the Beta Quadrant border.
I thought the navigator said something like, I have no idea where we are. They would still be able to see the same stars and pulsars. Either he is an atrocious navigator, or else they are much further away from home than the Klingon Homeworld.
That was my own take, obviously, but that was why I immediately thought "alternative universe." That, and the fact that they twice dropped hints that the spore "map" in Stamets head also included alternative universes. Quite willing to believe they are just playing with us though, especially after the "sleeper spy" business.