This is pretty easy with the Chromium Browser.1
As root, create the file
/usr/sbin/chromium-browser-session and fill it with the following:
while true; do chromium-browser; done
This opens the Chromium Browser in an endless loop, so it will reopen if somebody closes it.
Make the file from the previous step globally executable, i.e., execute
sudo chmod 755 /usr/sbin/chromium-browser-session
Again as root, create the file
/usr/share/xsessions/chromium-browser.desktop and fill it with the following:
Comment=Open the Chromium Browser (no window manager)
This creates an option in the X display manager to initiate the X session using the Chromium Browser instead of Unity (or GNOME, KDE, etc.).
Before logging in again, click the icon next to your user name and choose Chromium Browser from the drop-down menu.
Log in again.
Right-click the tabs bar and uncheck Use System Title Bar and Borders.ssion
Resize the browser window in all four directions, so it occupies the entire screen.
Right-click the tabs bar and check Use System Title Bar and Borders.
There is no window manager, so the greatest "damage" somebody will be able to do should be moving and/or resizing the browser window.
To log out, press Ctrl + Alt + F1, log in and execute
to kill the script launching the Chromium Browser.
1 It should be just as easy with Firefox, but I don't have it installed, so I can't test it.