I can vouch for this for sure. I'm 64 and I do bodyweight exercises now instead of lifting (but I also endorse lifting, especially for young folks, as long as it is free weights and you use proper technique).
If you're older, and not used to resistance training, no problem. Just start small and work your way up. It doesn't take long.
I bet one reason I don't have arthritis is because I do pushups. As Owen says, it's great for the bones, and I had never considered his explanation about what it does for the hands (although I guess I subconsciously knew).
I'll say this much - when I was young, I looked forward to my workouts. Today at 64, I do not, no matter how hard I try. I just don't enjoy them anymore.
But I love the results.
Oh, one more thing - downward dog push up for advanced users. Killer.