To answer your question about overtraining and injury separately:
Overtraining is putting your body through greater stress than it is equipped to handle. If you were talking about adding external weights to your pushups and chin-ups then you might hit that threshold. You can do pushups daily and not put your body through enough stress to overtrain it. However, if you are talking about hundreds of pushups throughout the day you might be pushing it.
Injury is caused by bad technique and muscle imbalance (which causes the former). It's important to make sure you exercise the whole body. You get the most bang for your buck with complex exercises that get a large number of muscle groups at the same time. So you may have a routine that you alternate each day. 5 minutes is enough to squeeze out a rep of 10 "whatevers" easily. You run the risk of letting your muscles get "cold" so be sure to warm up before you do your rep each time.
So as long as you are talking body weight, and you have specific goals of what you want to do throughout the day in your five minute breaks, it could work. You'll want your 5 minute routine to be something like this:
- Warm up for a minute
- Push out a rep of one or two exercises for three minutes
- Stretch (cooldown) for a minute.
You may want to pair a set of ab workouts (crunches, iso-ab plank, etc.) with your pushups/chin-ups.