Is it possible to recover data from a zero-out hard drive?

No, it is impossible to recover files from the zero-out drive given that all the drive data is overwritten.

Zero out data is an option when you erase a volume in Disk Utility on OS X Yosemite and earlier. It is also available to erase an AirPort Time Capsule in AirPort Utility on Mac. This option overwrites all of the disk's data with zeros. In a short period, Zero Out Data offers effective data protection. 

And Zero Out makes it impossible to recover data from the drive given that only the data that is not overwritten can be recovered with data recovery software or service. 

Normally, when people delete a file from the drive, the occupied space is marked as available, you can still recover them back with data recovery software such as iBoysoft Data Recovery for Mac, but if new data is stored in the marked space, the deleted data will be permanently erased, overwritten, then, data recovery is impossible.

If that is an internal hard drive on Mac, have you backed it up with Time Machine, or uploaded important files to iCloud Drive? If so, you can still recover some lost files with these available backups.