Сейчас у меня есть только Linux, есть ли способ загрузить мои электронные письма (включая отправленные) на мой жесткий диск в Linux (Ubuntu)?
Instead of using an email client, you might like to use an IMAP client such as imapsync, imapcopy or some similar tool.
This would allow you to copy from hotmail/outlook direct to mbox or maildir formats on Linux.
You could of course also do this manually using an email client and drag/drop :(
Yes. First you will need to get a client (like Thunderbird; there are other options as well) and then you will need to connect your client via pop3 or imap (each provider has different instructions for this - see here for a detailed walkthrough - https://support.mozilla.org/en-US/kb/thunderbird-and-hotmail
Basically, the process is 1) download the client you will use to access/download the email. 2) Get the details to connect to your mail provider via pop3 or imap and enable access if required (see last line of above link for more detail). 3) connect client to external mail account and let the data be downloaded.
Optionally, at this point you can backup your entire mail from the client to a compressed archive or similar.
The settings are a bit different for Outlook.com but are easily found.
In addition to the above, as the other answer says, there are options for just having a downloaded version of the mail and not access via a client which also work. Either of us could help with any questions regarding that solution as well. :)