A 208 Y 120 Volt Three Phase Four Wire service, be it from the Utility or your own Genset, can supply three legs (A, B, C) of single phase 120 VAC Line to Neutral power PLUS can be used to power a 208 volt three phase motor or other such 208 volt three phase loads.
SO TO ANSWER YOUR QUESTION YES a single phase motor (say 120 volt) can be ran off one leg of a 208 Y 120 Volt genset. It would be fed from any one of the legs (A or B or C) and Neutral, thats 120 volt single phase AC.
HOWEVER if your homes main distribution panel is the typical 120/240 volt Single Phase Three Wire, and you wanted to feed it via a Transfer Switch by a genset, you would ONLY be using two of the gensets three avaialble legs of 120 volt mind you !! PLUS it would only be 208 NOT 240 Line to Line. "Some" 240 volt equipment may not like that idea so well lol !!!!
NOTE if a genset had say 5 a KVA rating, you still only have total 5 KVA available, regardless if you were dividing the load up among the three legs.
NOTE while some 240 volt motors (depends on motor design and load etc) can operate at the lesser 208 volts, Id prefer a genset that supplies the same voltage as my utility provides and/or the load requires.
NOTE many commercial and light industrial services are indeed 208 Y 120 Volt Three Phase Four Wire, but thats NOT a typical household service
I guess what Im saying is if you have a 208 Y 120 Volt Three Phase Four Wire main service panel at your home, then a similar three phase genset and transfer switch would be practical HOWEVER if your serive is only 120/240 volt single phase three wire, then the backup genset should be similar.
YES its true if you have a 208 Y 120 Volt Three Phase Four Wire genset you can serve three legs of 120 volt (with respect to Neutral) versus only 2 legs if it were 120/240 single phase genset BUT WHAT ARE YOUR REAL NEEDS???
If you want to use 208 Y 120 three phase genset power for backup as your posting asks, HOW DO YOU INTEND TO USE AND DISTRIBUTE IT if everything you have is either 120 volt single phase or 240 volt (as typical for a home) ??????????
Unless you have some 208 volt three phase loads you need back up power for, I dont see why you want a 208 volt three phase genset (although yes it can deliver three legs of 120). Makes more sense to me to have a genset thats the SAME AS YOUR UTILITY FEED and use a Transfer Switch
ONE FINAL THING there is such thing as a 120/240 Volt Three Phase Four Wire center tapped Delta Red Leg Service (utility or as genset) but thats NOT for a typical household service. It can supply full true 240 Volt Three Phase Power,,,,,,,,,,,Three legs of full 240 volt Single Phase Power,,,,,,,Two legs of 120 VAC Single Phase off the center tapped transformer. Theres two legs of 120 volts single phase with respect to neutral, but on the red or wild high leg its 208 single phase with respect to neutral ITS NOT FOR THE TIMID OR AMATEUR.
BOTTOM LINE FOR ROOKIES 208 DOES NOT EQUAL 240 they are differently derived systems.
John T long retired Electrical Engineer but a guy retains these such basics