Most standard hotels in large cities or destination cities will have a gym.
You can pack the stretchy bands which are virtually weightless. Running is always available.
Shopping and finding a quality restaurant is sometimes a huge issue. Look online before you book your hotel.
And frankly, if you're the type of person who does body weight exercises anyway, you probably will do them on the road. So if you're not that type who is super self motivated, then you have to prepare and counter your bad habits with a plan to exercise and eat healthily.
The biggest stumbling block is booze. You're traveling on the company's dime, and what's 2 or 6 glasses of wine? Especially with clients. You wake up late, or tired, or hungover and aren't motivated to get into your gym clothes and go to the gym and squeeze in a workout before getting back, showering, getting ready, and starting your day of meetings and travel. And what are you going to eat that's clean and healthy for breakfast? When you're out of your home region and traveling for business, it's then you really come to realize how much crap food options are easily available, and convenient healthier options are damn near impossible to find.
Maybe pack healthy snacks like nuts and apples, or stock up after you've arrived.