I got stuck at the hairdresser's [and that's why I got home late] would be understood in context, but I don't recommend it, because “to get stuck” doesn't inherently convey the idea of getting “delayed” – it suggests nemohl jsem pokračovat dál, nemohl jsem se vymanit, uvízl jsem etc., so:
- I got stuck in the traffic (so I was late for work).
- (I couldn't finish the crossword because) I got stuck on the last three clues.
- (I didn't finish the exam in the allotted time because) I got stuck on the last question.
- (I started to panic when) I got stuck in the lift at work.
- (I started to write a novel last year but) I got stuck halfway through when I ran out of ideas.
In your context:
- I got/was held up at the hairdresser's (because they were short-staffed).
- I got/was delayed at the hairdresser's (because they had a power cut).
- I was kept waiting at the hairdresser's (because they were running late with their appointments). … nestíhali
- Milý deníčku… Jsem v hajzlu, uvízl jsem na Marsu… [Dear diary… I'm up shit creek. I've got / I'm stuck on Mars…]
- Nápad na sebe nechal čekat poměrně dlouho, nemohl jsem se vymanit z bludného kruhu. [The idea was a long time coming, relatively speaking, because I was / had got stuck in a vicious circle.]