If you have 5 GB storage, it is possible that it won't back up. Make sure you have enough iCloud storage to backup.
You can check how much storage is needed for the next backup by going to Settings> Name (Apple ID) > iCloud > Manage Storage> Backups > then choose the device you want to back up. There you can see the "Last Backup (Time of the last Backup), Backup Size (the size of your recent backup), and the Next Backup Size" (Here is the article from Apple)
If you see that the next backup of your phone is about 6GB (if you have 5GB plan), it won't let you backup your phone because you don't have enough storage left on your iCloud storage. However, you can subscribe to iCloud+ for additional storage and other services from iCloud+. In addition, if you have enough storage for the next backup, try to restart your iPhone or your Data Connection or WiFI.