In order to get a list of all (or part) of the mailboxes and view their sizes, simply use the following PS script:
Get-mailbox | Get-MailboxStatistics | Sort-Object TotalItemSize -Descending | ft DisplayName, ItemCount, TotalItemSize, Database -auto | Out-File c:\mbx_sizes.txt
The above command, will get all the mailboxes, and list their sizes and the database they are member or, then output the result to a text file called mbx_sizes.txt…
You can change the sorting to be Ascending, to have the smallest mailbox on top…
Also, you can get the sizes for mailboxes in a specific database by:
Get-mailbox -Database <DATABASE_NAME> | Get-MailboxStatistics | Sort-Object TotalItemSize -Descending | ft DisplayName, ItemCount, TotalItemSize, Database -auto | Out-File c:\mbx_sizes.txt
Replace <DATABASE_NAME> with the actual name of your database.
And of course you can do it for 1 mailbox as well by using:
Get-MailboxStatistics -Identity <USER_ID> | ft DisplayName, ItemCount, TotalItemSize