Let's say I have created a block with some Full HTML (Facebook Feed) text inside. How can I commit the block and its text content into my Git repository, so my co-workers can pull it?
That way, you can export block as a module, and modules are text files, so GIT accepts them with it's full support.
Then, there is Features to Git module. It claims:
Features to Git allows trusted users to write features directly to disk, commit them to git and push them to a remote. This makes it possible for site builders to be able to manage features in git without having to get their hands dirty on the command line.
Create a module for that block using the Features module and then commit that module on to the GIT repository of yours.
For more information you could read the README.txt of feature module from.