While on the road you don't have your handy scales at home, and without making an appointment to see a doctor, where else could you find some scales to determine your on-the-road weight? This is useful if say, your travel medicine is weight-related...
Ask at your hotel or hostel. I work in a hostel and we have about four sets of scales so guests can make sure their luggage won't be over the weight limit for their flight.
I'm sure we're not unique in this regard.
When you are at the airport you could step onto the conveyer belt of a checkin desk that is not in use. The scales normally still show the weight.
If you happen to find a shop that sells stones (for stone carving) they usually have scales to measure the weight of the stone to calculate the price. Those scales should be able to hold you.
You could try to befriend someone local and ask to borrow his or her scale. You could also use the couchsurfing website to contact people and ask them just to borrow their scale.
Or you could try to find a store that sells scales and just use it in the store without buying it.
Sometimes you can find a scale inside a gym or swimming pool (maybe more common in Scandinavia?)
In many countries you actually have people standing around on street corners with their scales which you can use for a small fee.
The problem may be that you may use your home scale naked, while you are dressed to various degrees when on the street (depending on the weather). Also some of the scales look pretty old, and may not be super correct. But at least is it gives you some idea.
In the UK pharmacies sometimes have scales that require a coin to be activated.
If your hotel room has a bath, measure the dimensions, fill it up completely with water, climb in and measure how much water is displaced. Don't blame me if the hotel complains about water damage :)
For completeness sake, pharmacies in Italy often have both scales and sphygmomanometers to measure blood pressure. Both are available to the customers, and are either free or coin-activated.