Loving all the Pet Shop of Horrors love in this thread! That thing deserves more recognition. I need to finish the manga someday...
Anyway, here's some:
Iria: Zeiram the Animation
The title is due to this apparently being a prequel to a pair of live-action movies (which I've never seen). A bounty hunter woman tracks a powerful alien across the galaxy for six episodes, plus there's also some political intrigue.
Vintage promo.
Mermaid's Scar
A one-off OVA based on a chapter from Rumiko Takahashi's
Mermaid Saga manga (which is also recommended). DO NOT get this confused with the Mermaid Forest TV series (same source material but worse execution). This is one of those sad cases where there was never a DVD release and I've never heard of a subtitled version (that wasn't HK-tier).
This youtube upload is the entire VHS tape, including commercials at the beginning.
Gall Force: Eternal Story
If you can live with some cheesecake-y elements, this is a rather interesting sci-fi story that... goes places. The basic premise is it follows the crew of a ship during the final hours of an intergalactic war, and then weird things start to happen. The first two sequels are also good but have that "unnecessary" feel to them.
Trailer (this is for the first three movies)
Bubblegum Crisis
This is one of those cases where "obscure" is relative: fans of anime from way back when will know this one, but its probably forgotten in this day and age. Nobody has ever been able to decipher what the title is about--the basic premise is that its the future and this mercenary team called the Knight Sabers (women in power armor) are investigating crimes related to robots called Boomers. It has sort of a Robocop-ish feel to it.
There were sequel OVAs (Bubblegum Crash and AD Police Files) and some TV series. I can't speak to the quality of all of those though.
Best promo I found for this one was
this one for the modern bluray release.
Ruin Explorers
A four-episode fantasy OVA that is obviously trying to be Slayers (itself a series I recommend but I wasn't sure if it counted as "obscure") but still has enough charm to hold its own.. and one of the characters is a dog who I swear is Muttley from Wacky Races.
Old VHS promo
Hikaru no Go
So this one has a fun story behind why it exists: the creators of the original manga (one of whom was actually a pro Go player) noticed that manga and anime would get kids to want to play stuff like Yu-Gi-Oh or Beyblade... why not do the same thing for a traditional board game?
The result is one of the best shonen series ever, and one barely anyone talks about. Though some aspects of its concept are clearly inspired by Yu-Gi-Oh (kid gets into a game after awakening a spirit), the direction is very different, telling more of a slice of life character growth story.
My one regret is this is one of those cases where modern HD versions tend to have shit subtitles. Nyaa still hosts a torrent of the old fansub by Elite-Fansubs, which is what I recommend if you can live with the visuals being not-HD.
(Couldn't find a promo for this one).
another way to find classics
Back in the day there were all sorts of ads and promos on VHS and DVD, in fact I have a volume of Devil Hunter Yohko that came with an "ADV catalogue" that's almost nothing but trailers, and sometimes I watch that just to remind myself of when anime used to be good.
You can still find a lot of them on Youtube just by searching things like "AD Vision promo" or "US Manga Corps promo" or the like, and in the day this was the primary way we learned about anime.
I used to be a huge weeb and trying to come up with items for this list actually sparked nostalgia, so I may come back with more later.