If email were architected to put the cost of storage on senders then I think the spam problem would have been more manageable. Instead it's so closely modeled on physical mail that recipients get overwhelmed quickly.
Even if email worked like that, I don't think it would solve the spam problem. Spam emails are rarely personalized and the same exact message is sent out to many different addresses.
As such, the spammer would only have to store each unique message and the list of recipients associated with each unique message.
Hrm. I wonder if this is an actual use case for blockchain and proof of work.
If you want to send me an email, you have to send me an SMTPCoin. If my user doesn't mark it as spam, I give you the coin back. If my user marks it as spam, I keep the coin.
This was the inspirstion for nano's use of proof of work too. They don't use PoW for the chain itself but do require every transaction to include a small proof of work to avoid spamming the network, almost identical to hashcash.