The thing is, the history of that era is huge and has many parts. The BBS Documentary covers some of it, but IMO it has a lot of flaws and underwhelming segments given the subject matter. I personally know a few people who were in that video and I can say there's nothing inaccurate or bad about what they said, but I don't feel like Jason really "got" it that well. It's better than nothing and I appreciate it was made, but it's a bit ambitious and thus needs to gloss over things.
I'd probably look into reading about some of the following which relates some to the different parts of the BBS era:
* BBS Software w/ interesting stories - Clark Development, Wildcat, Renegade, Telegard, WWIV, Oblivion/2, Forum...
* ANSI Scene - ACID, ICE mainly, then the smaller groups like Dark, Fire, etc.
* Demo Scene - Future Crew, TRSi, etc.
* Warez Scene - Razor1911, PWA, DoD, Pentagram, TRSi, THG, Fairlight, Paradox
* Music Scene - Trackers, MODs, S3M, XM, music groups, artists (Purple Motion, Doctor Awesome, etc.), Sound cards (AdLib, Roland, Sound Blaster), MIDI
* HAM Radio WRT to BBSs
* BBSs and the Internet - Mail gateways, usenet, IRC
* Phreaking - Captain Crunch, War Dialing, Boxes (blue, red, etc.), and more...
* Cracking - Mostly overlap with warez
* Emulation - Overlap with warez a lot, but there was a pretty interesting ROM dumping scene and early NES emulation scene starting
* Alternative Lit/Anarchy/Etc - Spreading this kind of stuff was really popular and most of it is now "illegal" per many laws in western countries. Most famous is Anarchist's Cookbook, the first version.
* Early Internet and versions thereof
* IRC, Usenet, Gopher, Archie, FTP, etc. is all interesting as some of those things rose and fell during these times
* Programming Languages - C, C++, Basic, Pascal, Smalltalk, Delphi, VB for a range of where some of the industry was going or didn't go. Obviously many more.
* Platforms/OS - IBM, Tandy, Amiga, C64, Atari ST, CP/M, Unix, HP/UX, Windows, Dos, DRDos, OS/2, OS/2 Warp, BeOs (later), Novell Netware, Apple II/Mac, Sparc, SGI (Irix), Desqview (used to run multi-node BBS sometimes) and many more.
* Law Enforcement Operations - Raids (ex: Steve Jackson Games), Busts, etc.
The above list is not meant to be comprehensive, just to guide you what articles, books, videos, and such that might turn up if you search for some of these things. It's a long road.
Regarding Halt and Catch Fire, I think there are things about it that ring true, but it's not so accurate. People get things done way too fast, act too much like we do today, and the office culture feels off to me. They really don't capture some of the more innovative cultures at various big companies of the time. Not even the more crazy ones like Atari's old office, though there are clear nods to companies if you pay attention. It'd be a long post to dissect it, but the short answer is no, but it's not like I experienced all there was to experience. It feels to me more like they took a lot of real things at the time, composited them together for story purposes, put on some 80s clothes, and otherwise threw things out the window. I do enjoy the show anyway though for even daring to touch that material.
Anyway, I hope that gets you started on a long, interesting path and I could add tons of more bullets to that list.