To install the GNOME desktop environment on a Gentoo system, you'll need to follow these steps:

First, make sure your portage tree is up to date.

   sudo emerge --sync
   sudo emerge -u world

You will need to install the base system and then add the GNOME packages.

   sudo emerge @gnome-base/gnome-base-devel

Ensure you have a basic X11 setup. If you don't already have an X server installed, you can install one like `Xorg`.

   sudo emerge xorg-server

Now, install the GNOME desktop environment along with other necessary components.

   sudo emerge @gnome-desktop

If you don't have a default display manager installed, you can choose one like `gdm`, `lightdm`, or `sddm`. For this example, we'll use `gdm`.

   sudo emerge gdm

Edit your `/etc/conf.d/xdm` file to set up GNOME as the display manager.

   echo "DISPLAYMANAGER=\"gdm\"" | sudo tee -a /etc/conf.d/xdm

Enable and start the GNOME Display Manager.

   sudo rc-update add gdm default
   sudo rc-service gdm start