Most of the email services have policies that allow them to close your account under a variety of rules, and of course occasionally a service simply disappears. In most such cases, unless you've physically and intentionally created backup files on your own computer, all your old emails will be lost.

As a particularly bad example, Hotmail automatically terminates your account if you fail to log in for 30 days. (Don't go on tour or take a long vacation.) Once terminated, your old email is permanently gone. There is apparently a "grace period" of 30 days(?) during which you can renew the account to keep your same email address, but the deleted messages cannot be restored.

With a free Hotmail account, attempting to send the same message to ten or more recipients within the same 24 hours usually will result in "suspension" of your account on the assumption that you might be sending "spam." Repeated "suspensions" may result in closing your account (without explanation) which will result in loss of all prior email messages - and your address book. ("Extra pay" Hotmail accounts come in different flavors, and a less subject(?) to arbitrary termination - maybe.)

I would expect that most email services have similar policies. As long as your email is on their server, they can make it inaccessible to you. This would likely apply to gmail, and it is strongly recommended that everyone should check with their own email service to know and understand how reliable and secure storage of messages on the servers owned by your email provider actually is if you're not making regular backups on your own machine.

John