You don't want to do that. Trust me on that one.
I did this last year as I had a difficult time with people and was getting way to upset, so told myself I would try to just not care anymore. It worked, to the extent that I now am struggling to care about stuff and people that are really important to me, have lost interest in getting to know new people (and I don't care anymore), and will most likely need counselling to fix this. I am a little less unhappy than I was overly feely, but *not* caring has brought with it a set of problems of their own. Both situations are not good to be in and are causing me heartache.
Try to tone it down, become less sensitive (maybe seek out a counsellor; if you're depressed this may be an idea anyway), but don't go down the route I did.