So many websites have tried to compete with eBay, and so many have failed. That's because they're not just trying to compete with eBay, they're trying to be eBay. But eBay is on top of its game, and without a very large budget or an incredible amount of time to devote to marketing, it would be very difficult to aspire to the level of sales this behemoth company enjoys.
Not to say that you can't compete with eBay, only that you should not try to be eBay – or any other large company, for that matter. The key to success is finding your niche within the market. Take a look at Etsy, and eBay-like auction interface where buyers and sellers meet to trade. Sounds a lot like eBay, right? Except Etsy's niche is in handmade wares. Because sellers only sell handmade items, Etsy has a well-defined niche that includes a very personal touch to merchandise purchased.
This isn't to say eBay can't lose its market share, either. When it changed its terms and conditions, some sellers (and buyers) became quite perturbed. Some were so distraught that they ventured to competitors like Bonanzle. These sellers are diversifying so all their eggs aren't in one basket.
So if you want to be the next eBay, Amazon, Twitter, MySpace or whatever, you have to identify your niche within that market. Break down the target audience to find sub-target audiences. Do it again. And again, until you find something truly unique you can take to market. Consider: eBay users → that want to buy from people, not companies → that like hand-crafted items. Niche. Etsy.
Or: eBay users → who do not like the terms and services → but don't have an alternative. Niche. Bonanzle.
Here's one that I'm not sure has been done yet: Twitter users → that shop online → that use coupons. Another niche – hey, maybe you can make the next Twitter, only this time customers tweet about online coupons instead of just anything that comes to mind.
How about some brainstorming? Take one of the big online companies and break its audience down into a sub-niche and tell us the results!