The diameter of the biggest part, namely the camera head, is 23mm/0.9in. 

I have not looked for a specific travel case, but unless you are being really rough with it I would say a padded envelope or a generic small case would do it. It's pretty sturdy. I take it to to the office sometimes and just put it in one of the pockets of my laptop case. Also, for extra protection against scratches I never removed the clear plastic film that comes over the screen. Except for the red ""pull here"" tab, you can't tell it's there. 

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I don't know the exact size but it even fits 12-13 year old kids. 

Yes, it can be used without a phone.