Try the answer to the FAQ Sweet Home 3D hangs when I try to save my home. How can I fix this?

I can access Open and save when I run the app in Chrome browser (but then I can't access the menus).

Saving and opening homes in the online version doesn't work the same way and has nothing to do with the File > Open / Save menu items of the browser. This can be done only for registered users with the dedicated buttons in the toolbar of the applet.