I don't know of anywhere that offers such a service. Unless you're into reading technical books, in which case O'Reilly have an online option :P
Considering it takes a second or two tops to download a complete book though, I have everything in the "archive" on amazon (although I keep most on my eink kindle too), both books I've bought with them and those I've emailed to them. Then i can download what i want to read whenever I fancy a change if I'm out somewhere and don't have the book on my current device.
The kindle app is on most devices and platforms now so it's perhaps the closest you'll get. Unless you have more than 5GB (I think that's the limit) of non-amazon bought ebooks, in which case it's not going to offer enough.