Is it impossible to store 2^266 records on a database of some kind?
Absolutely possible, because someone with the resources to store 
6021188640340442162025691220451771208370039202613309330051794368412 Terabyte
 definitely has enough money to get a custom made database system for his purposes as well.

I might be available as a contractor, for only 0.01€ per Terabyte.