Is it possible to install and run both gnome and kde on Ubuntu and enjoy the best of both worlds without having to switch between them?
That depends what specifically you mean.
Of course, you can install both environments, the packages themselves are not mutually exclusive. You can start with one environment, and you will be able to run the programs from the other. For example, runnning KDE programs with Gnome is not a problem.
However, only one environment would be responsible for actually managing your desktop.
you can .
after you must login
it will be other desktop with ctrl+alt+f8 if you want first you will press ctrl+alt+f7