Yes, that part of the film is accurate. The Bundesbeauftragte für die Stasi-Unterlagen (Federal Commision for Stasi Records/BStU) maintains a central Berlin office and twelve regional offices for the preservation of Stasi files. The newsmagazine DW has a good video on the BStU archival work. The BStU offices allow for individuals to gain access to the preserved Stasi files and reports.

Access though to these files is not open. German Basic Law provides citizens the rights of a private individual. So there are certain metrics and barriers to gain access to BStU files. The BStU website outlines both the criterion needed for access as well as pdfs of the request forms. The archivists than evaluate and either approve or reject these requests. The Lives of Others's Georg Dreyman would have access to his file because the legislation that erected the BStU allowed the subjects of Stasi surveillance to see their files. He would not though likely be granted the right to see his girlfriend's file as federal law preserves her right to be a private citizen. Mark Reeder, a musician who was active in the late GDR music scene, recalled in the Cold War Conversations podcast that he still cannot gain complete access to his file because of these regulations. Established researchers can view Stasi files but privacy rights still remain very iron-clad. So hypothetically, if we defictionalized The Lives of Others, as a researchers we would have to use a pseudonym for Dreyman in our subsequent writings.

The film is a pretty accurate dramatization of the process of reading the files, although it does telescope the bureaucratic process of application and approval for dramatic purposes. The film does not illustrate the fact that the files are curated by archivists who block out components of the file that are not directly pertinent to the applicant. But even with these precautions the files can still be rather large. Timothy Garton Ash's The File describes the process of reading his file:

Meanwhile, every page of every document you get to see has been renumbered by the Authority’s archivists, with a neat rubber stamp over the Stasi’s own careful handwritten pagination. It’s like a parody of German thoroughness. One extreme follows another. Probably no dictatorship in modern history has had such an extensive and fanatically thorough secret police as East Germany did. No democracy in modern history has done more to expose the legacy of the preceding dictatorship than the new Germany has.

A special law, passed by the parliament of united Germany in 1991, carefully regulates how the files can be used. Frau Schulz {the BStU archivist Ash works with} has read my file before I did because, in scrupulous bureaucratic implementation of that law, she is supposed to photocopy the pages on which the names of Stasi victims or innocent third parties appear, to black out those names on the copies and then to copy the pages. gain, just to make quite sure the name cannot be deciphered using a strong light. She is also meant to cover up any passage containing personal information about other people that is not directly related to the inquiry. But what is not relevant to understanding a secret police that worked precisely by collecting and exploiting the most intimate details of private life?

These files often contain not only the names of direct informers (IMs) and what they said, but also minutia and massively irrelevant details. Some of these Stasi reports can be inaccurate. Ash's book details how some of the reports on furniture do not align with his memories. The British reporter John Green recalled in the Cold War Conversations podcast that the Stasi reported that he and his wife used classic evasion tactics on a drive in the country when the mundane truth is that they got lost and had to make a sudden turn to get back on their route. Other reports show obfuscation among IMs who report on minutia and gossip.

All of this underscores the absurd obscenity of the Stasi as a secret police. On one hand, it was most efficient at gathering up as much information as possible given the technology of this era. But most of this information was garbage and irrelevant to the Stasi's self-proclaimed mission of state security. Knowing Katarina Witt's sexual liaisons might stimulate prurient interest, but they do little else useful from an intelligence standpoint (young people have sex, who knew?). The Stasi had mountains of data on many East Germans, but the agency missed the mounting anger at the system building in the 1980s that culminated in the mass protests which helped topple the regime. The common theme that emerges from people who have read their Stasi files is not only a sense of betrayal, but also befuddlement as to why they were the subjects of surveillance to begin with.