ACCESS YOUR MAILS DURING INTERNET DOWNTIME

Various email clients are available, which users can use during the Internet downtime. One has to sync all his important accounts to access the mails when their is no internet. 

There are several Extensions / Add-ons available in browsers like chrome,safari,etc.,  

For example emails on one of the famous web-based mail client GMAIL can be accessed offline on Chrome using an extension called Gmail Offline. 

A short description regarding gmail offline extension.

Gmail Offline beta is a Gmail app built to support offline access, allowing mail to be read, responded to, searched and archived without network access. After first start-up, Gmail Offline will automatically synchronize messages and queued actions anytime Chrome is running and an Internet connection is available. More than just an offline app, the streamlined user interface, based off Gmail’s popular tablet interface, provides extremely fast response time with a pure, email focused experience.

To start Offline Gmail after installing, open a new tab in Chrome;  In the new tab pane you will see a Gmail Offline Icon. Click on the icon, and Offline Gmail will load.

Gmail automatically sets a date range depending on how much mail you have in your account to avoid caching several GB of data.

To choose manually how much recent mail and which labels Gmail offline keeps on your computer:

Follow the Settings link in Gmail. Go to the Offline tab.Pick how much back you want to have all mail available offline under Recent message range: (under Download options).Click Change Label Settings to pick a different time-frame. Choose a maximum file size for attachments to keep offline under Specify maximum attachment size.Click Save Changes.

Try searching for such add-ons/extensions even for different mail clients.