That sounds like you have a very limited online mailbox. Most mailboxes have been 5GB or more for years now, so must be an older ISP type mailbox or they limit the size if it’s a free acct.
You can setup eg: Auto Archiving in eM Client to download messages older than X days to a local Archive folder within eM Client or if you have eM Client V10 you can now also Create a Data file as per @Gary post on your hard-disk and move older messages and any older messages folders into that to then clear space from your online mailbox.
First though I would contact your mailbox provider and ask them if they can increase your mailbox. Ask what would a eg: 5GB or 10GB etc cost a month, as could be they have many options you can do to then keep messages all stored online.