Stain on fibreglass Eames chair  

The dark lines are probably created by some sort of deeply embedded mold or dirt in the threads, and from what you say it seems pretty clear that scrubbing and sanding isn't going to get it.
I know it will sound awful to some, but in cases like this I have sometimes had different degrees of success with a very carefully applied bleach-water solution. (Since the color in the shell is in the resin and not a coating, it does not bleach out the color very easily, if at all in most cases.)
If you put some very tiny clumps of wet paper towel saturated with bleach-water on very small areas and leave them there for a few minutes before rinsing, the bleach can sink into the threads/cracks and take the color out of whatever dark matter is in the threads-- without discoloring the resin. You might have to experiment a bit with different strength solutions and also different amounts of time, and repeat several times.
I have found the shells to be quite good at holding up to a wide variety of strong cleaners and even some solvents, as long as everything used is non-abrasive. In this kind of case, I have found the bleach water method to work far better than even some of the very strong solvents.
Again, the bleach water strength and the time factor are both very important variables. Go slow, raising the stakes carefully and incrementally, and rinse very well after anything you do, and you should be okay.
Hope this helps. These look like great shells!

BMartin
You may use generic spray oven cleaner with a nylon nail brush for difficult stains. This will not harm the fiberglas and it is impossible to create esthetic damage. You have nothing to loose in the chairs current condition.

Might be a good candidate for trying a poultice. Mix a paste of unscented baby powder and laundry bleach or hydrogen peroxide (hair-bleaching strength) to the consistency of peanut butter. Trowel onto the stain about a half-inch thick overlapping by an inch or so, cover it with plastic food wrap and tape the edges down with blue painter's tape. Allow to dry completely (48-72 hrs). It's the drying process that helps pull out the stain. Brush or vacuum off the residue and rinse thoroughly.
There are a number of poultice recipes for different types of stains and repeated application is sometimes necessary. Don't mix ammonia and bleach!

Did you end up trying anything? I have an extremely similar looking side shell (the stain is in a different spot, though) and I would love to try getting the stain out, without sanding, if possible.

That is a wicked spot for a stain, but at least not many folk would dare sit on the sweet shell. Not me.
Try hydrogen peroxide, looks like blood.
Best,
Aunt Mark
ps I'm usually wrong.