This can be an annoying and time consuming issue, and it's one with two parts.
The first and most common question is what size seat post do I need for this vintage frame? If you have bought an old frame on it's own, it can be anyone's guess what size you will need as there is so much variation. On steel bikes, the shape of the tube entry can make it awkward to measure (you can try with calipers).
Sometimes the old seatpost has the size engraved on it, but if you are working with a frame without any components the quick way to get an accurate measurement is with a seat post sizing gauge.
All bikes shops will have one of these or something similar, or if you have to buy your own this one is made by Cyclo Tools and cost around £15-20. The gauge simply drops into the seat tube and is marked to tell you the size.
The second issue is when you have a seatpost, but don't know it's size. Either you want to sell it on, reuse it on another bike or replace it. If you are lucky it will have the size engraved on it. It's worth giving an old seatpost a very thorough clean as feint markings can be revealed under old grease and grime. If there are no markings then it's a case of measuring then confirming.
To measure with calipers. Hold them flat and level. Most seatposts are in 0.2mm increments. However, you do get the odd curveball (such as the Birmalux 26.3mm seatpin or even worse the Sturmey-Archer 25.48mm). Don't squeeze the calipers too tightly, slowly rotate the seat post to get an even measurement. This measurement is easier if you have a vague idea of what the post size will be, for example a lot 1980's lower end Raleigh frames use the chrome 25.4mm seatposts like this one.
If the measurement seems to keep changing on a vintage seatpost then it's likely your post is damaged and out of shape. Steel posts in particular can become oval shaped from years of persistent over tightening. Measure a few times higher and lower on the post and you should get a good feel for the size.
To confirm the measurement:
Get a piece of paper that you know has straight edges, cut it into a strip approx. 10cm x 3cm. Cut from the corner so you retain two straight edges. Wrap it around the post but make sure you line up the straight (non cut) edges squarely. Draw a short straight line as thinly as you can, on the exact edge where the paper overlaps itself.
Measure line with your calipers then divide it by Pi to give you the diameter. In this example the measurement is 79.80mm which divided by 3.1415926536 = 25.4011289174mm.
You will probably never get a 100% exact match as the thickness of your line will affect the measurement, but if the measurement with paper is within 0.1mm of your first measurement then it is most likely correct.