No, I'm afraid you've misunderstood what "Archive" means in the context of Gmail.
"Archive" simply means "remove the 'Inbox' label" from a message/conversation. The original conceit of Gmail was that you really don't need to delete anything; you just need good search tools to find your old messages. At the time Gmail launched you got an unbelievable amount of space (at least, compared to other web mail services at the time).
Gmail Help: Archive messages in Gmail
To free up space in Gmail, you would need to actually delete messages. You could possibly use a third-party client and IMAP to move the messages to another mailbox so that you don't lose them, but that's a whole 'nother topic.
There are some other questions (and answers) here which you might find helpful:
Of course, you should check to ensure that it's Gmail that's using your space: