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The HTML of most blocks is controlled by a "template" that is stored in the database. To ensure that you cannot mess up things totally and have no way to roll back, the system has the "default template set" which is not possible to edit.
If you want to edit the template for blocks or module pages, you will first need to clone the template set in System Admin -> Templates and then set this new template set to be the one used - this is done in System Admin -> Preferences -> General Settings.
After these steps, you will be able to edit the block template from "edit block"