I'm not a native speaker of German, but I have been learning German all my life, including spending two years at college in Germany, so I think I have enough Sprachgefühl to understand the problem and intuit an answer.
"The Estate" as a group in The Strange is interested in two things (see for example http://www.montecookgames.com/earth-and-recursions/):
- exploring recursions
- keeping the earth safe from monsters who creep in from recursions
To my mind, this means that the meaning is closer to meaning 1. of "Estate" on Wikitionary, namely "(now rare, archaic) State; condition." The Estate is interested in understanding the state of recursions, and preserving the current (safe) state of Earth, of preserving the status quo. The fact this meaning is rare and archaic is part of what makes it a cool name for a clandestine organisation.
For the German rendering of this, my choice would be "Der Status" (or possibly just "Status" without the article). This has both the possible meanings of "die Lage, die Situation" (http://www.duden.de/rechtschreibung/Status), and also hints at "Status quo" without unnecessarily evoking the rock band. Also to my non-native ears it has something of the mystery of the unusual use of "The Estate", being a Latin loanword. I suppose you could go with "die Lage", "die Situation", but they seem too bland to me.
A bonus of "Der Status" is also that, even if you think the meaning of "The Estate" is more about social status or organization (as per mxyzplk's answer), then Der Status also contains the meaning of ("Stand, Stellung in der Gesellschaft, innerhalb einer Gruppe" - Duden as above).