This is a very frequent request and there are many threads on this. Some of these threads do include some workarounds but use caution. Even when you use these there are some ways that people can access some items. Anything that someone sees they can copy although not always through ancestry's copying methods. There is a difference between items for ancestry's copying method. Some items can be viewed but not copied through their method. If it is really private don't post it.
Otherwise here are some options from least secure to most:
1. Public tree not attached to anyone living
2. Public tree attached to someone living
3. Private tree not attached to someone living
4. Private tree attached to someone living with invitees
5. Private tree attached to someone living with no invitees
You can chose to attach any photo to a living person. You can also attach a photo to more than one person and make one person you attach it to private (living). This becomes very complex but also provides different levels depending who it is attached to and how many.An extra word of caution with this because these may be viewable to someone for whatever access of privacy you have for that profile and there are clever people here who can probably copy those, although not using ancestry provided methods.
Once something has been public it will probably show up public no matter what you do.
The one thing you cannot do is block a particular person from viewing or copying an item. I realize this exists on other sites but I do not believe there is any workaround for this here other than a private tree with control over your invites.
None of this is foolproof so if it is really private don't post it.