What eBay needs is competition.
True, and the problem is eBay has critical mass. Everyone knows about them and they're the de-facto place to sell your stuff.
Which poses a problem for a new site that wants to compete - they need to get word out, and they need to attract buyers and sellers. Attracting sellers is easy - you just make it so your fees are a lot lower than eBay (not hard). Attracting buyers, though, is a lot harder.
First, buyers know what eBay pricing is about, and if they're coming to your site, they need a hook. One common hook is cheaper prices - why pay $100 for this item on eBay when you can pay $75 on our site! That attracts buyers quite easily.
However, then you have the opposite problem - sellers know what they get on eBay, and they want eBay pricing because well, why should they get $75 from your site when they can get $100 from eBay? Only if the eBay fees make it so they get less that way than through you would they want to go through you.
And buyers, no they won't pay eBay prices on non-eBay sites because they made the effort to come - if they're going to pay eBay pricing, they'll return to eBay.
Perhaps you'll try the payment angle. The fact is, if you're Joe Random selling stuff, Paypal is the only way to go in the internet era. Otherwise you're going to have to accept money orders sent through the mail, which is a huge PITA and a general annoyance Why would anyone bid on an item, then line up at the post office to buy a money order then a stamp to post it? That's so ... 80's mail order. Buyers want to whip out their credit cards, click "pay" and enter those details in, and boom, payment done in a few seconds, not hours. So maybe you'll try being the middleman, which might work (if you can deal with all the payment card stuff).
So the crux of it is - your new site must sell stuff below eBay's prices or buyers won't come, it must give sellers more money than eBay would, and it must make it as frictionless as possible with easy payment systems designed for modern ecommerce.
And I've heard plenty of buyers who complained about alternative sites - not worth visiting because it's the same prices as eBay. And I've heard sellers complain buyers lowball them - they get more money from eBay. So the deck is stacked against you - buyers want to pay less, sellers want at least the same.