It depends.
I'm not a DIY-guy at all: I can put together something from Ikea, change a light bulb, but that is about where it ends. If I need to paint, install equipment in the kitchen, do sanitary works, etc. then I happily pay the professional to do it properly for me.
About flat car tires:the one time I had one I was in a rental car in Italy, they have garages everywhere (because the roads suck) and for 10€ I got it fixed. The rental company told me that i needed to change the tire first but I did not know how to do that... Even fixing a tyre from a bicycle is something that I would rather do by the pro.
But other things I like to fix myself. In particular electronics: I opened several desktops, laptops and phones before to fix them.