The town finds peace about Barb's death -- through a lie
The funniest thing about the Barb plot in Season 2 is how much it echoes the arc of Season 1: Again, Barb has a larger role in the beginning of the season but pretty much gets forgotten as the tension rises in later episodes. In Episode 5, Nancy and Jonathan pay a visit to the investigator hired by Barb's parents Murray Bauman, played with scenery-chewing glee by comedian Brett Gelman, and they share their tape with him. Intrigued, he suggests they "water it down" by claiming that Barb was "exposed to some dangerous toxins." He argues that the real story of the Upside Down and the Demogorgon is simply too radical. Instead, they need something "scary but familiar."
So, they send the tapes out with a letter to a bunch of newspapers and then… nothing really happens with the Barb plot for the next few episodes. Most of the cast members are busy trying to figure out what's wrong with Will; Eleven is hanging out with punks in Chicago; and Nancy and Jonathan end up sleeping together. At the end of the Season 2 finale, after Eleven has closed the gate and Will has been freed of whatever demon was possessing him, we get a "One Month Later" title card and learn from some newscaster voice-over that the release of the "incendiary tape" has thrust Hawkins into the national spotlight.
That's right: the "justice" for Barb is mostly delivered in an expository info-dump. "Under mounting pressure, several high-ranking members from the US Department of Energy have admitted involvement in the death and cover up of Hawkins resident Barbara Holland who died due to exposure to an experimental chemical asphyxiant which had leaked from the grounds of the lab," says the voice on the news. We see images of Barb's parents at a funeral for their daughter -- and with that, the show is done with Barb.
In a way, it's a fitting end to the Barb phenomenon: Barb cult-dom was clearly motivated by the feeling that she was shafted by the show's creators, served up as monster meat to a plot that would quickly focus on other more conventional TV characters. If the first season had dealt with her death in a more serious manner, the fandom around her wouldn't have had the same touch of advocacy to it, which is what made it so effective as a meme.
Season 2 doesn't really provide justice or closure for Barb -- her death is still part of a cover-up -- but it does pay tribute to essence of the Barb meme. It was never really about her. It was about the people who loved her.