M Fan, always glad to see a person ask questions and try and do things right. You asked for it so here are my comments: Wait n see, electrical or legal questions usually bring out the most opinions of any topics here lol and I enjoy and love reading every one.........
A) "I'm thinking of running 8 AWG stranded from the main box in each barn to the outlet location (probably 60-80 feet run)."
I did this in a hurry so no warranty but for
8 AWG Copper,,,,,,,,,240 Volt Single Phase Motor,,,,,,,,,,,80 Ft run,,,,,,,,,,,26 Amps,,,,,,,,Looks like the voltage drop would be 2.686 volts or 1.12% which is acceptable and its something Id definitely (if my calculations are correct???) do versus using only 10 Gauge. But do as you please, I'm ONLY here to offer help and suggestions, but NOT tell others what to do.
B) "I'd have a 30 amp shutoff switch by outlet. From the outlet I would use a heavier extension cord, 10 gauge, out to elevator motor."
Id be sure the 30 amp switch is BOTH in sight of and within 50 feet of the elevator. Our shop always used lockable switches so the electrician could padlock the switch when he was working downstream of it. That's kind of how they are made as a safety precaution.
C) "From the outlet I would use a heavier extension cord, 10 gauge, out to elevator motor. "
Okay here's the deal and dont anyone have a calf yet: TRUE 10 Gauge wire (subject to insulation and enclosure and temp etc etc) is generally rated for 30 amps.............TRUE 26 amps is less then 30.............TRUE (Billy Bob) 10 gauge will work and suffice fine. I just have to say, however, when I was an AC Power Distribution Design Engineer I perhaps "over engineered" or was "over safe" or "over conservative" lol and I didn't prefer (when and where possible) to have a continuous load of over 80% of a conductors rated ampacity i.e. I didn't prefer to have over 24 amps continuous in 10 gauge wire BUT THATS JUST ME....... (But see below regarding true ampacity
HOWEVER if you look deep at wire ampacity charts the insulation class and enclosure and ambient temperature and jacketing all contribute to the safe rating, and if used outdoors in free air the ampacity can be higher then if the conductors are say enclosed in conduit or buried or in free air etc etc etc.
D) You mentioned a "motor starter" I take it theres proper thermal overload protection for the motor ???
E) Any overcurrent protection for the conductors FROM the switch TO the elevator will be sized, of course, to match the wires ampacity. That could be fuses or circuit breakers (TWO POLE OF COURSE) such as a two pole fused safety switch (Id use like Class KF Dual Element Time Delay fuses to allow motor start up if I recall correctly, been a while sorry) versus a non fused safety switch if that's the overcurrent protection location. If not already so equipped with proper thermal overload protection, you know they make Combination Starters which suffice for BOTH the safety disconnect PLUS have the starter contactor and heaters/thermals all built in to a single unit.
F) As I'm sure you're already well aware that 240 volt (Only 240) single phase branch circuit requires the two Hot UnGrounded Phase Conductors L1 & L2, PLUS the safety Equipment GroundING Conductor which IS VERY IMPORTANT in that outdoor metal device application !!!!!!
Best wishes, keep safe with outdoor 240 volt equipment, the life you save may be yours or your family.........