How do I backup my Mac without a different storage device or computer?
Solution 1:
Saving two copies of your files to the same drive isn’t a backup type that will protect you from a Boot Camp error.
You’ll want to get an inexpensive USB drive to copy your critical documents. Then when you can afford a drive that’s at least as large as your internal drive, set up Time Machine so you are protected by automatic backups of the Mac data. Note, Time Machine does not backup Boot Camp.
Solution 2:
The problem is not Bootcamp per se. The problem is that in order to install another operating system on your Mac, you need to have a dedicated partition for it. And in order to get that, you have to shrink the existing partition on your drive to make room for another one, and that shrinking can irreversibly damage your data. Disk Utility is pretty good about trying to do this intelligently and not include your data in the parts that get removed when the partition is shrunk but it isn't perfect.
So it usually goes fine, but you're rolling the dice which is why Bootcamp gives you that warning. So the problem with your idea is this: where's that backup going to be at? Because if it's going to be on your primary partition it's subject to the same potential corruption when the shrinkage occurs.
So backing your data up on the same drive is out. If you're willing to potentially lose your data forever then just roll the dice and take your chances, and if you're not you can buy a very reasonably-sized external drive to use for Time Machine for a fraction of the cost of a Mac/iPhone.