In SharePoint online, you don't have the ability to use run these commands the Get-SPDatabase or Get-SPContentDatabase as they are not available. You also don't have the ability to access SQL Server to check the size of content database.
Your first choice is to use a tool/webservice to download the entire site collection and then check the size. This is not the best use of any given tool or worth writing a service for.
This is the best way to get the size of the content. Site Settings->Storage Metrics and you could get data like below and using simple Math you could get fairly accurate estimate of the size of content.
% of Parent Size * X = Total Size
75.27/100*X = 181.5
i.e 181.5 * 100/75.27 = (Total Size of Content)
