Find Linux options here: http://alternativeto.net/software/sandboxie/?platform=linux - maybe someone can pick up https://github.com/tsgates/mbox
At least 1 VM is strongly recommended but you could containerize within that.
Edit: toolchain for running a browser instance in a chroot jail. With a head. GUI. On the desktop - it's here today (with commercial support), just run Windows+Sandboxie in a VM. Yowch!