With some effort, you can meet your needs using web sourced emails. I do it with my ISP, gmail and Outlook.com emails now. Thunderbird makes it easy to do so. Also, Thunderbird has a local folders feature that you can move your web based emails to as needed. The full-fledged Outlook program does it by setting up and opening pst files.

The advantage of this is your normal emails stay on the web within folders that you can control. That means they are always accessible if you lose access to your main computer. If you decide to drop your web based email account, just use Thunderbird to move them to your local folders. Same if you need to clear up space to stay within size limits. With Thunderbird (or the full-fledged Outlook and I'm sure many others), you have an off-line copy of every email that is also on the web. I like that as it gives me some assurance that I won't lose my mail store through a single point failure.

Since I went to this type of system, I no longer have to deal with the fact that an email is trapped on a computer that I don't have with me. I can get to it from my phone, tablet, or any other computer with internet access.