C."Sorry, but there's no way that that cable is rated to safely carry 300 amps."
"How do you know that?"
"Well, for one thing, I can bend it."
Split over 2 pairs, it's 2.5A per conductor. Assuming 75C-rated insulation, you need ~25 AWG - call it 24 gauge for a little safety margin. That's a conductor with cross-sectional area of 0.2mm^2, diameter a touch over half a millimeter.
It's not a human hair - but it does intuitively feel like too much.
We'll probably get used to it. And only see houses go up in flames once in a while because some cheap non-UL-rated cable got purchased from a shady dealer...
(I know Matt doesn't need to be told any of this, it's just for the thread...)
Oh, and a wire rated to carry that many amps from the electrical utility to your home will need a cross-sectional area of two million circular mils of pure copper.
Yes, it's a crazy traditional unit. In more modern terms, it's over a thousand square millimeters. Good luck!