How can I tell how full an Azure Storage account is?

You have to enable Storage Analytics. Then read about Storage Account Monitoring. Finally a look at the Analytics Metrics table(s). Note that it takes minimum 15 minutes until metrics are updated.

And btw, Azure Storage account limit is 500 TB as per Azure Subscription and Service Limits.

UPDATE

After reconsidering, the only way you can get the full storage capacity report is via the Account Billing page. Simply click on your name on top right (management portal v.Current, a.k.a. https://manage.windowsazure.com/) then chose "View My Bill" option.

This page is updated on a daily basis.

There is no API or other programmatic way to get these statistics.

Now it's possible to get it in Azure Monitor. In Azure Portal navigate to All services -> Monitor, click Expore Metrics and select your storage account. There are different useful metrics and Capacity metric is among them.