I have a S8 and was wondering if it is possible to make my number appear as Private...
On a per-call basis, you can prefxx the number dialed with *67.
There are definitely options to make your number appear as private on Caller ID of others. We have a free feature called Share Name ID where you can add a name (or the word "Private"), so when you call people who don't have your number saved in their phone, your number doesn't come up, and it comes up whatever you set it to. Check it out here: http://spr.ly/6606J72zg
Verizon user here. Just FYI in case the same happens to you. Years ago I marked my number to appear private, and have never been able to switch it back despite several different VZ agents' attempts to help resolve it. The workaround (in case you need it) is to store the default number prefaced with *821 ("1" being the country code for the USA/Canada).