Web-based email accounts tend to have the same capacity for all folders (Sent, Inbox, Deleted, etc). If it's a 1Mb Inbox, then the total is 1Mb.
Transferring them to computer will save space as Outlook Express (as with most programs) will store them on your hard disk (the "memory" you describe). However, they must be deleted off the internet account to take advantage of this. For example, I have a Hotmail account. If I download my messages, they will remain on my Hotmail account until I move (or copy and delete) them into my Local Folders.
Your computer can store as much as it has room for, and there's no limit other than the space on your disk. If you have 1Gb spare, then that's 1,048,576Kb, which if the average email is 3Kb long, then that's 349,525 emails.