As an Exchange Online Office 365 Global Admin you have the ability to review EXO365 MBX sizes for your users and set send/receive limits, which effectively set the MBX size. To learn more, please check out the following Knowledge Base (KB) article: http://support.microsoft.com/kb/2490230
Except from article:
To configure an Exchange Online mailbox, use one of the following methods.
Configure mailbox size limits for a single user
- Connect to Exchange Online by using Remote PowerShell. For information about how to do this, visit the following Microsoft website:
- Run the following command to set the mailbox size for a single user:
Set-Mailbox <User ID> -ProhibitSendQuota <Value> -ProhibitSendReceiveQuota <Value> -IssueWarningQuota <Value>