In principle, with secure deletion programs.

Leave me the disk with the compromised information and I'll get my hammer drill. A good steel drill bit and you'll see how long it takes me to make it completely impossible to recover anything. A dozen holes in the plates and I assure you no one will recover anything.

Apart from those cited by Nestor in the previous answer, the safest thing to do would be to destroy the storage medium on which the information is stored. And by destroy, I mean obliterate it completely. If it's a hard disk, uncover it, dismantle it, break and burn the platters and then leave the remains abandoned in the sun.

You can also format the drive and overwrite all the storage multiple times with tools that exist.