The "Make Startup Disk" ubuntu utility sees it as writable only if it's fat formatted. If I format it to Ext4, it sees it as 0 free space and doesn't allow to click "Make startup disk" button.
However, since I plan to use the drive to run the OS (not just install) and will store files bigger than 4gb i would like to have bootable usb drive on ext4 Would that be possible?
Yes. You can do what you intend.
The correct approach to your desired result is to simply "Install Ubuntu" to the external drive rather than use "Make Startup Disk" utility. That utility is designed to create a small boot disk as a method for troubleshooting and repair or installation, not for running the OS as you intend.
If you want the "Try Ubuntu" Option on the drive as well, you can simply dd the ISO image to the drive as outlined here: https://help.ubuntu.com/community/Installation/FromUSBStick#Install_and_run_Startup_Disk_Creator_alias_usb-creator
I think you're dealing with a permission problem.
Use:
sudo chmod -R a+rwX,o-w /media/[your_drive_name]
sudo chown -R $USER:$USER /media/[your_drive_name]
Then eject the drive and insert again. Now try to create a Startup disk.
Or just boot up a live disc and install Ubuntu to the USB drive instead of the internal HDD.
