This article accompanies the fable
Beethoven lived and worked in Vienna for 35 years and he moved constantly - he may have lived in at least 60 places during those years.
In 1802 he was living in Heiligenstadt, then a rural village in one of the wine-growing areas (heurigen) and now in the northern suburbs of Vienna. It is where he wrote the "Heiligenstädter Testament," a letter to his brothers lamenting the onset of his deafness and really a last will and testament. We don't know if it was ever sent since it turned up with the Immortal Beloved letters in 1827 after Beethoven's death.
The house he is most likely to have lived in at the time is shown at bottom. It is now a museum (one of several in Vienna dedicated to him - this website here has links to many of them).