Generally speaking, anything above 4GB will do the job. Some Linux variants may require larger volumes, so pay attention to the requirements when downloading.
When you're looking to create a bootable Linux USB drive on a Mac, the first step is to make sure you've got the right USB drive for the job, and that it's formatted correctly to avoid any problems. You can go the freeware route for an easy option, or put a little bit of time into creating the drive yourself using Terminal.
There are a few ways to create an Ubuntu (or other Linux) bootable USB drive for Mac. Linux has long been synonymous with bootable flash drives, whether it's to fix some sort of problem with your primary operating system, or for trying various distros.