Not sure what email program you are using. Presuming it's outlook:-
Go to File from the menu - select 'New' and then select the Personal Folders File (.pst) option.
Select the location you want to save the file in from the drop down option. This can be either a drive on your machine or on a network if you are at work and local drives are disabled etc. Next, click 'Create'.
Enter a name for your folder and add a password if you wish to secure it (you will be prompted for a password everytime you want to read mail in that folder but is useful if you store it on a server for example), and click ok.
From your normal folder list view menu (ie. Inbox, Outbox, Sent Items etc), you will now see a new folder (with the name you chose when creating) which you can use for storage. It will act like any normal 'online' folder except won't be using space on your email account. Emails within this folder can be viewed when email is not functioning etc.
Hope this works for you. I believe personal storage folders become unstable once they reach about 2gb in size. If it becomes that big I would suggest creating another folder and begin using that.