(Not trying to imply that vellum should be the only storage method but that sounds like cheap robustness, and I'd not want to throw that away just because you want to be capital-D Digital.)
Their current pricing is $0.0007 per GB-month, or $0.000064 per KB for 800 years in today's dollars. Since the Latin text of Magna Carta is 28 KB of UTF-8 text, that works out to $0.0018 to store the text for 800 years. You'd obviously also want a high-resolution scan of the original document. I don't know how big that would be. If it was a 100MB image, it would cost $6.66 to store for 800 years at Amazon's current Glacier prices.
Obviously we're making immense assumptions about the reliability of the service, of price stability to deliver the quality of service, and so on. And perhaps the biggest issue with this analysis is that we're also relying on huge economies of scale for it to be profitable to deliver an archival service like this over centuries.
If this sounds crazy cheap to you, you're not alone. As technologists we're used to dealing with huge volumes of data, so it's easy to forget that historically these volumes simply did not exist.
Even assuming the corporate entity will survive that long is dicey. To ensure survival of a document on such as service you can not just look at the cost of a storage service, but the cost of infrastructure to care for it. E.g. just the matter of keeping an organization alive for the duration to ensure an account is kept active and live and prepared to ensure each and every monthly bill is paid.
It's a social not a technological problem.
But how to do it is much easier to solve.
Why to do it is the actual problem. What to keep and when to keep it without knowing the future.
How does anyone know that the 100 photos that a person from Morocco put onto Flickr 5 years ago would have value in the future? Why should those be archived? In 200 years these photos could be immensely valuable.
How can this potential future value ensure the archiving of the photos today? Is there economics for this?
Not to mention supporting the UK's tourism brand... castles and queens and whatnot.
The Situation Room is part of the Whitehouse's brand whether tourists get to see it or not.
Not saying it doesn't help tourism, its just not the intent of it all & doesn't have much bearing on the decisions being taken as much as the simple act of continuing traditions.