After further consideration and removing my first answer, I now note that you don't want to see the readable text in a binary (e.g. with the
strings utility), but see text in binary form.
So, I think the KDE utility, okteta is just what you want, and it is available in the repositories and can be installed with
sudo apt-get install okteta
It allows you to view text files in binary form (see the second screenshot below), and you can click the tab at the bottom of the page to switch between binary form, hexadecimal, decimal, and octal. You can even create a new file and start entering text and, if you have the binary mode selected, the characters typed will be shown in binary, just like in the online converter you linked to. The screenshot directly below shows an example of this:
This application does exactly what you have specified in the question, and it is a gui as well, so that seems to tick all the boxes.
Okteta seems to be the most fully featured editor available, and there are also modules and plugins that give additional functionality.