Are you sure you didn't just accidentally check out with everything in your cart? From what you wrote, it seems the "system messing up" was on your side (pebkac). It's much easier to blame someone else though.
No, the site was riddled with 500 errors, their CDN was down which made all the fonts not load, etc. They had it on their Twitter that they were having a weird system outage after I realised and was sure after I tweeted them they would refund the domains. No dice. 100% refusal to refund.
That should have been a good hint to not use Godaddy