As an user it would be great if there was any mechanism in place to prevent pushes to Bitbucket if the repository is bigger than a specified size.

As a suggestion, add to the repository settings (which can be controlled only by the admins) a specific size which the repository cannot go beyond. In case that size is reached, the commit is halt and an approval to the current repository administrators is requested. Only if the administrator approves this commit, it will be merged. Discard the changes if the commit is not approved within a specific amount of days.