T20 doesn't have a No Career Class. The closest you can come to that is rolling up your character with any Skills & Feats you get from growing up on your Home World and then going out into the universe as a level 1 Character with no Class. Those Skills are considered Class Skills and can be advanced accordingly while any other Skills are considered Cross Class Skills unless you use Hobby Feat to get some or from Feats, like getting Gunnery from Ships Weaponry Feat. You wouldn't have access to any Class Feats. You still get your starting Feats and Feats at every level divisible by 3.
Besides, a level 1 Character isn't really a no class character. They've had their initial training and instead of doing a Prior History to level them up, you are leveling them up from Level 1 through role playing. Not easy but possible.
Although this sounds like a cool challenge for me, it's also possible for a Character like this to Cross Class into one of the Core Classes if you meet the requirements. Or the reverse, things got so bad that you can't continue in any Career and are basically gaining experience as an unemployed/homeless/vagrant type of Character, hoping for a reversal of fortune or maybe thinking there is no way out of the darkness.
Thanks for the idea
@GypsyComet !