Earlier quoted context omitted.
ether = aether = "a medium that was once supposed to fill all space and to support the propagation of electromagnetic waves". This definitely suggests that what their trying to communicate here is that they don't want to see their names circulating on mediums like the Internet or other oldschool broadcasting mediums like newspapers, radio and television.
In Swedish, "rymden" kan also refer to just the "space" or "attention" really. It doesn't imply any public space really, just any space.
One way to look at the use of "space" there is that an English speaker would say "room", as in "chat room".