By default, the local mailbox size is 1GB when using G Suite Sync for Outlook. You will receive a message about your Local mailbox is full once it reaches this limit. You can increase the size of your local mailbox by doing the following.

Find the Google Sync Tool in the Tray. Click on the ^ icon and find the Icon highlighted below (Green Circle looking icon) then right click on it and choose Set Mailbox Size Limit.

You will then receive this dialog box to change the size.

You can set it to Unlimited and it will download all of your mail. As the dialog box indicates you can always check your mail on the web to do a quick search of all older email.