On suitable volume, it should be possible to offer a 0.1% edge and be profitable (over the long term - as variance can cause even a site with a 10% edge to lose money in the short term).
My impression is that the gambling websites with a 1% edge are making large profits.
Just-Dice.com has a 1% house edge, but after more than 1.1
billion bets has only kept 0.32% of the amount wagered. The reason not many sites offer much less than 1% edge games is that the smaller the house edge, the greater the effect of variance, and the more likely the site is to go bust. In other words, the smaller the edge, the bigger the bankroll the house requires to be safe from going bust.
Sites which offer player-vs-player games can obviously afford to take a much smaller cut, since they are guaranteed to win a little from every bet.