I have a 1980 Fuji America 18 speed road bike that needs to have the handlebar replaced. How do I determine the handlebar size? It appears that 26mm is the common size for older road bikes - but I do not have a micrometer to be able to take an accurate measurement.
If you don't have good measuring tools (and know how to use them, but that's not too hard) it will be very hard to tell the difference between 25.4 mm and 26 mm – until you try to fit the bars in, then it will be obvious…
For a bike of that vintage odds are good (according to my recollection and Sheldon Brown's Crib Sheet) that the size is 25.4 mm.
So what you should do is by a set of 25.4 mm bars (ideally from a nice LBS that you want to support anyway) and see if they fit. If they don't and you can't return them get a 0.3 mm shim – Nitto and I think some other make them. If you're feeling flush get it with the bars just in case.