Okay, heres the deal, but this is how it was when I was an electrical power distribution design engineer and things may have since changed so no warranty.
SOME equipment is labeled and designed to operate at EITHER 208 OR 240 while some IS NOT, so the correct engineering answer is IT DEPENDS. That being said, much of equipment will still operate at EITHER even if not so labeled perhaps not as efficient or as much output if operated at a reduced voltage.........
The ONLY time I specified 208 Y 120 volt three phase four wire was for say an office complex (NOT an industrial application) where the majority of the load was 120 volt single phase but the HVAC was operated at 208 volt three phase with compressors and air handlers designed for use at 208 and clearly labeled as such.
The more typical industrial service was 480 Y 277 three phase four wire in which case I specified 277 volt lighting and used a dry transformer to get the needed 120 volt single phase
As some correctly stated, on the 208 Y 120 volt three phase four wires system its 120 volt from any leg to Neutral (3 legs of 120) but 208 from any line to other line since 120 x square root of 3 (1.732) = 208.
The thing is I see ALOT MORE 240 volt three phase equipment then I do 208 since 240 is more the standard then 208 although sure, "much" equipment can still work at reduced voltage
If I had my druthers and I needed a lot of three phase 240 (more typical equipment) instead of 208 plus still some 120 Id opt for maybe the 120/240 volt three phase four wire High/Red leg Delta system.
THE CORRECT ENGINEERIGN AMSWER DEPENDS ON THE LOADS AND USES AND HOW MUCH 3 VERSUS SINGLE PHASE EQUIPMENT YOU WILL USE and that cant be answered without first knowing all that. Sure Billy Bob and Bubba can render an opinion or what worked best for their shop/home but that may or may NOT be best for your individual situation
NOTE I do like the idea of three phase but 208 isn't my first choice of voltage, that would be 240 for small shop/farm use.....
Your money your choice, CHECK WITH THE UTILITY PROVIDER and do a little design and leg work as to your loads and uses and the equipment youre gonna operate is my engineering advice ......