[Electro Shock Blues] is definately their/E's masterpeice, but if you listen to it first, you will definately assume that Eels are emo. Which is more than totally untrue.
Emo? Nah... Emotional, and lots of that, but I think emo has been applied to pretty much everything by now, most of it whiny, thirteen-yearold bullplop.
ESB has pretty much the best buildup I have ever heard. If you know the backstory (basically, E is the last living member of his entire family), you'll undoubtedly feel how the album was an immediate reaction to all the chaos that's sure to erupt in a guy's mind at times like the ones he found himself in. Yet, it doesn't have any kind of "I'll cry forever and ever and ever and ever and ev..."-kind of mood. Rather, it's reassuring, uplifting and pleasant (provided you listen to the whole thing, without skipping songs- some find
Hospital Food too messy/noisy).
Shellac has been mentioned. So do it.
Other stuff I can recommend would be:
-Jeff Buckley (He's pretty straightforward, but delve into his stuff and hopefully, you'll discover the same richness of spirit and musical brilliance that I and many others have)
-Kira And The Kindred Spirits (rock n' roll/garage/blues/soul... I guess)
-Mew
-Kashmir (Okay, this is more alt-rock. than indie, but brilliantacular, I promise)
-Interpol, of course. Goes without saying, so I'm typing it instead.
-Swans
-Naked Lunch
-Joy Zipper, possibly...
...And while you're at it, get yourself hooked on Tom Waits, though there's not much indie in that guy.