Dropbox has a web interface.
What ever happened to FTP servers?
ssh ftp is the best bet imo, get a dedicated server with raid and linux and you've got yourself a storage box with high bandwidth capacity that's in the "cloud" (internet = cloud in this case). You can also automate stuff, and you can even setup a local mapped drive using sshfs if you want to.
except most people have asymmetric pipes.
Cloud computing does not magically give you more bandwidth. You will still be constrained by it.