The important first step for media support in core just landed in Drupal 8.4.x: a new beta experimental Media module to support storing media of various types. While Drupal core already has generic file upload and image upload support, the new module will support asset reuse and be extensible to support video, remote media, embedding, and so on.
This is a huge testament to individuals and organizations with shared interests pulling together, figuring out how to make it happen in core, and getting it done. 89 individuals (both volunteering their own time and from various companies all across the world) contributed both directly in the core patch and via involvement with the contributed Media Entity module:
That said, this is just the first step. (If you go and enable the core Media module, all it can do right now is give you an error message that no media types can be created.) The next steps are to add a file/document media plugin and an image media plugin so these types of media may be created on the site with the module. Then, widgets and formatters for the upload field and image field interfaces will be added so we can reproduce the existing core functionality with media. Adam Hoenich wrote up a concise summary of the next steps, and granular details are listed in the followup roadmap.
There are definitely more tasks than people available, so your contributions would be more than welcome! Now is the time to make sure media is integrated in a way that your projects can best utilize it. Get involved through the media IRC meetings happening at 2pm GMT every week in #drupal-media. (See https://www.drupal.org/irc for more information on Drupal IRC). Or, if you are available at other times, ask in the channel. The issues are listed on the Media Initiative plan.
Let's put the remaining pieces in place together!