Usually it does in areas that don't get exposed to steam/heat - like around stoves, sinks, or dishwashers. Contact paper works really well inside cabinets, when it is used for shelf paper.
As Fiddle stated, it will if it's not exposed to steam/heat and also if you're sure you stick it firmly down. Another thing is to make sure that your cabinets are completely clean with no grease or dirt on them before you start. It won't last for years and years, but it should last for a decent amount of time. Now, if you want to do the inside, that's similar but different because the edges are constantly "scraped" or "nicked" when putting dishes away. That can mess up the edge. I like to use the Rubbermaid (or other brand) matting that comes on rolls on the inside of my cabinets because it stays put well and I can take it out to wash it if I want. I personally, would rather paint on the outside of my cabinets because I don't have the patience to be meticulous enough to do it right. :( I hope you do. :) Wishing you the best.
If you can swing it, a hometalk project used the following stuff and she says it works great as a cabinet cover - https://www.hometalk.com/42284153/kitchen-make-over-without-paint-for-few-dollars
It won't last forever, but if carefully applied and not exposed to lots of water, I have personally scraped it off a cabinet it had been applied to years ago, and it wasn't easy getting it off. I think what helped, was that my mother had wrapped it over the top and bottom of the door so there were no seams on the exposed edges.
Not in my house, but if you rarely cook and clean, it will last a while. Just know, removing it will be more trouble than it's worth. I would seriously consider painting before using contact paper. Just my opinion as I cook several times a day.
Amazon carries a heavy duty contact paper for doing cabinets, and I also see many use the Peel n Stick Removable Wallpaper.
It works best inside of cabinets on the shelves where it doesn't get wet