In general, everything will stick around as long as you paid for it. Your email account and your web site will probably be unceremoniously deleted once the money runs out, and the domain name itself will be freed up. While you use your domain to point to your web site and your email account, they won’t be associated forever: Once they expire, the name is available for someone else to fill it with new content, not to get access to your existing content.
Oooooor: You could bequeath all your online assets to a family member in your will (don’t forget to give them all relevant passwords), possibly along with some money to keep paying for the domain registration, email- and web hosting.