That was an ad for a pizza restaurant a few years ago... made me laugh.
I tend to go for the cheap insurance. For any pole, the advantage of another foot of depth is cheap for the benefits you attain in strength and outright brutishness. I would just get your regular post hole digger and go for it, at least 2 feet deep.
As far as the clotheslines themselves, you will need to find a way to tension them easily. Screw tensioners work well, and if you are a creative engineer, you could probably devise a neat way to tension all of your lines with one screw. However, if you are more simple minded like myself, a simple turnbuckle on each line will do nicely. Make sure the clothesline and parts you use will not rust on the clothes, absorb soil or droppings that will be leached into your clothes, etc. You might consider a way to cover all or part of this, depending on how often it rains around your house, (forget this part if you live in a certain part of texas).