Skypig,
Several months ago my wife thought the battery was dead after listening to the radio for about half hour and it would not start. I cleaned the battery terminals and it started and ran fine since. I suspect there was a slight film or corrosion that was not readily visible, but prevented the battery from getting a full charge from the alternator when running. That may be a possible cause in your case - hard to know the history since you just got it. I doubt the starter with so few miles. I would more suspect the battery because of the age factor.
Before springing for a tune up at 27K miles, I would try Redline gas treatment (Pep Boys in NLR has it) or Techron fuel system treatment. That should clean up the fuel system deposits from relatively low use of the vehicle.
If the previous owner used a "cheap" regular gas (or even premium), there may be some deposits based on minimal fuel additives to keep the engine clean.
Congrats on a really nice low mileage find.