Sharepoint - How to add Newsfeed webpart to SharePoint 2013?

Yes. You need to activate Site Feed feature in Site Settings --> Manage site features. Then, when adding web part, under Site Collaboration group you will have the Site Feed WebPart

UPDATED

Yo must activate this feature in _layouts/15/ManageFeatures.aspx

As stated by others: The "Site feed" feature needs to be activated on the site where you are trying to add the Newsfeed webpart.

But there is one very important gotcha: The "Site feed" feature also needs to be activated on the site collection root site if you are working on a sub-site. I found this out by accident a while ago, went back and checked it now on SP 2013 SP1, and the issue has not been fixed. You still have to activate the feature on the root site.

Just a clarification - This is NOT an issue but just the way SP is designed.

Essentially allowing for more granular control across your site.

Features need to be enabled top down (Application all the way to the subsite level) Only then the feature is available. Still needs to be turned on - does NOT get turned on by default on all sites (understandably as you would not want features turning on by default across the few hundred sites/subsites you may have).