Twitter "stories", called Fleets, allow you to post content that disappears after 24 hours. But if you want you can mute some people's fleets.
Unfortunately, there is currently no way to disable all Fleeds on Twitter en masse. Your only option is to mute them manually - from individual accounts one by one.
You can mute the Fleets of someone who has made a recent post. So go to the Fleets at the top feed on Twitter.
To mute a Fleet, press and hold the Fleet of one Profile to reveal hidden settings. Click "Mute @ (Twitter account)" where the "Twitter Account" will enter the profile you want to mute.
To remove the mute someone, visit his Twitter profile, then click "Unmute" next to the message "You Have Muted Fleets From This Account".
Twitter does not remove the space it has created for Fleets, even if you mute Fleets by everyone accounts that you follow, so unfortunately it will continue to take up space in your feed.