Q:

Share the clipboard between bash and X11

In this thread the top answer shows how to copy text that has been previously selected with the mouse on a gnome-terminal, to the clipboard in X11.
My question is: Say I copy a piece of text from the terminal using bash set-mark and copy keyboard shortcuts (i.e. set-mark + M-w). Is it possible to share this clipboard with X11? 
EDIT: In the original question, I talked about sharing the clipboard with GNOME, but as Gilles points out below, GNOME doesn't specifically have a clipboard (it's general to X), so I have updated the question. 

A:

Bash's clipboard is internal to bash, bash doesn't connect to the X server.
What you could do is change the meaning of M-w to copy the selection to the X clipboard¹ in addition to bash's internal clipboard. However bash's integration is pretty loose, and I don't think there's a way to access the region information or the clipboard from bash code. You can make a key binding to copy the whole line to the X clipboard.²
if [[ -n $DISPLAY ]]; then
  copy_line_to_x_clipboard () {
    printf %s "$READLINE_LINE" | xsel -ib
  }
  bind -x '"\eW": copy_line_to_x_clipboard'
fi

If you want to do fancy things in the shell, switch to zsh, which (amongst other advantages) has far better integration between the line editor and the scripting language.
if [[ -n $DISPLAY ]]; then
  x-copy-region-as-kill () {
    zle copy-region-as-kill
    print -rn -- "$CUTBUFFER" | xsel -ib
  }
  x-kill-region () {
    zle kill-region
    print -rn -- "$CUTBUFFER" | xsel -ib
  }
  zle -N x-copy-region-as-kill
  zle -N x-kill-region
  bindkey '\C-w' x-kill-region
  bindkey '\ew' x-copy-region-as-kill
fi

¹ 
Gnome doesn't specifically have a clipboard, this is general to X.

² 
As of bash 4.1, there is a bug in the key parsing code: key sequences bound with bind -x may not be more than two characters long. I think bash 4.2 fixes some cases of longer prefixes but not all of them; I haven't researched the details.

A:

@Gilles already gave an excellent answer. I would just like to mention the existence of xclip, which is also a very useful way to copy terminal output to the X clipboard, by just piping anything into it:
$ cat /etc/passwd | xclip

