How can I reduce the Ubuntu install size to the minimum, with GUI?

Solution 1:

From the Ubuntu wiki sytem requirements page: Xubuntu seems to be as close to your demands when you want a GUI:

• GUI.
• 2 Gb. (5GB+ as of 2019)
• has a browser.

Lightweight GUI alternative (Xubuntu)

If you have an old or low-spec computer or want to get the most out of your hardware, using a lightweight desktop system such as Xubuntu is recommended, as it should make more efficient use of your system's resources. Of course, even if you have the newest equipment out, you could still use Xubuntu.

If your system has less than 192MB of system memory, use the Alternate Installation CD.

Note: If you have a low-specification computer, certain features may be automatically turned off to conserve system resources. For example, if you have a graphics card with only a small amount of video memory (VRAM), the boot-up screen may not be shown.

Follow this link for detailed instructions: Installation/LowMemorySystems.

Minimum system requirements for Xubuntu would fall roughly between Ubuntu Server and Desktop:

256MB of system memory (RAM)
2GB of disk space
Graphics card and monitor capable of 800x600 resolution 

Regarding processors, a recent thread on the Ubuntu Forums describes a 500 MHz Xubuntu machine as being "too slow" and discusses alternatives: http://ubuntuforums.org/showthread.php?t=1267443

Alternatives but not getting closer to your pre-requisites so it would require manual labour from your part:

• server install. 1 Gb required but no GUI and no GUI browser.
• minimal CD install: 20 Mb required.

You can download the i386 here: Ubuntu 11.04 "Natty Narwhal" Minimal CD (others you can find at the minimal CD install link above.

To install, boot your computer from the the Minimal CD and type "cli" (command line
install) at the prompt. You can then follow the instructions from the text-based 
installer. After the base system is installed, log in, and type "sudo tasksel" 
to select the system to install. 

Instead of tasksel you can do a sudo apt-get install chromium-browser on command line. apt-get will figure out what needs to be installed and will install the minimal required software to get chromium running (replace with firefox if you want that or with a lighter browser to save some more space ;) )

Regarding browsers: a very light one is Ephipany. So

sudo apt-get instal epiphany-browser

instead of chromium-browser might be better.

Solution 2:

Last I checked, the normal desktop install clocked in under 2gb.