Is it possible to keep my saves for Fallout 4 on the PS4 cloud? I don't want to risk losing my saves due to hardware error, or other error.
Yes, you can upload your savegames from any PS4 game to the Cloud if you have a valid PS Plus subscription and enough space left to upload. Fallout 4 is fortunately no exception.
Sony also recently increased their Cloud storage capacity from 1 GB per system to 10 GB per system. In other words, you can upload 10 GB worth of save data for PS4 games, 10 GB for PS3 games, and 10 GB for PS Vita games, for a total of 30 GB across all PlayStation systems.
Considering that a Fallout 4 save game is approximately 5 MB in size (on PC), you could upload about 2000 different save files for Fallout 4 to the Cloud, so space shouldn't be an issue any time soon.
Just don't forget to save the game in the first place. Your PS4 won't be able to upload your victory against a Super Mutant Behemoth, if the save file doesn't exist in the first place, after all. Don't forget to configure your PS4 properly, so it can auto-upload those files too, unless you enjoy manually uploading them instead.
As stated already, you can most definitely save saves to the cloud, provided you have a PS Plus subscription.
However, just keep in mind that if you're talking about saving "Survival mode" game saves, you are only allowed to keep 3 created saves at a time (per character). This is due to the fact that you're only allowed three "autosave" saves per character, and that Survival mode only allows you to make autosaves (through the "sleep" function).
In addition, you're also allowed an exit save (which is erased once you continue that save) and the rare hard save that is automatically made by the game at certain points in the game (for example, directly before leaving Vault 111, and midway through the "Curtain Call" quest).
I only mention this because it wasn't something obvious to me (I just assumed all of my saves were being saved in storage), and it wasn't until a ways in that I realized all of my earlier saves had been perpetually written over, with only my three most current saves being available.
All that said, assuming you have PS Plus, enough free storage on your PS4, and an external hard drive, you can feasibly keep up to 9 different save files at a time (per character), with 3 on PS Plus, 3 on the PS4, and 3 on the external hard drive. Not very elegant or practical (especially with Survival mode being the most difficult setting and much easier to die on than the other settings), but it's something, at least.
I can't get Fallout to recognize the cloud saves, in fact I tried to download the cloud save using the PS4 app in settings. It says that the game is not installed when it clearly is and works. But cloud won't download it... I borrowed a copy for the first 25 hours, then got my own copy and that copy works, I can start a new game, but it does not see the saves.
So, no, Fallout 4 does NOT work using cloud saves. Perhaps each disc has its own serial number or something, and if so, f Bethesda... I haven't had any issues with any other PS4; I regularly borrow games, and sometimes by my own copy afterwards and those saves work.