Others have pointed out how to access e-mails still on your provider's servers. As an alternative to Old_Geezer's suggestion for accessing your downloaded and stored e-mails (e.g. on your own PC), you could take with you a portable hard drive onto which you have transferred your e-mails. In fact I carry around with me in this way all my files and simply boot up from the hard drive wherever I am because it includes an operating systems - every computer I use is thus as my own and all content/work is purely for my eyes only, always. This can only work when/if you can freely boot up the computer you are using and attach peripherals at will, perhaps supervised. At holiday resorts, internet cafés, etc. you are unlikely to be allowed the freedom you need - but if/when you can do this it permanently solves the matter of accessing your e-mails and other files.