In the case of Wuala, they actually were totally available via web browsers [1]; they used Java, however, instead of Flash, as seemingly their desktop client was already written in Java, so this was convenient. (Additionally, Java came pre-installed on most systems, albeit Windows shipped with the ancient 1.1 version, of course, due to that lawsuit.) You can find presentations from 2009 discussing the technical issues that went into this [2].
[1]: http://www.wuala.com/en/launch/
[2]: http://jazoon.com/portals/0/Content/ArchivWebsite/jazoon.com...
This method did, of course, involve getting pulled out of the web browser, but AFAIK that was not a technical requirement of Java for this purpose. (They claim in the presentation that a "Web App", even with Flash, wasn't capable of client-side encryption, which doesn't make much sense to me; maybe they just didn't want to implement the encryption algorithm? I also do not see them considering embedding the Applet into the website as opposed to just using it to launch their external app, but again: convenience to their existing codebase is likely a strong factor.)
Well, actually - it only seems to work fully on Chrome.
The better version of this post likely would have been a "state of browser-based encryption"; we're so close to really being there but any possible implementation is still a slow hack that won't work on all systems.