Frankly speaking, the best is to use combination of both which is IMAPS. With IMAPS the connection is secuie (IMAP over SSL, or rather, TLS), and the client retrieves the list of emails and when you click on email, it is downloaded. Similarly Exchange protocol works this way. So it keeps email on the server so you can browse them online and downloads the one you need (or you can make some clients download it anyway). Searches are also performed on the server (in Outlook this happens if you choose to not store email locally at all).
To fully download emails, you'd use POP3S (POP3 over TLS).
However, in todays world it's the cloud holding our data and not devices.
10MB limit on email is small, modern email systems can handle big email mailboxes no issues. They are sometimes compressed and indexed, so holding 100GB mailbox is not an issue.
When you download emails, you need to backup your whole system or outlook PST file. This adds additional hassle plus you need to make sure your PST file is backed up securely.
Downloading emails is very risky, you should today use web interface with 2FA (Two Factor Authentication). This way Evil Sysadmins or Cleaners won't download all your emails by either reading your harddrive or getting Gmail web account password from browser saved passwords.
Since stealing emails is such big risk and issue today, you'd be good with encrypting the harddrive. It's in the MacOS, Windows (BitLocker) and Linux (LUKS) if you really want to download them.
Some harddrives also support encryption, so you can use it as well (in BIOS), and the backup PST file to encrypted USB stick. To encrypt USB stick you can use Bitlocker.