It goes without saying they're a luxury (or non essential or however you want to frame it).
However, whether you include the cost in your budget depends on your overall situation and what other non essentials you're paying for.
If you're in debt, have no savings, not saving for a pension, need to pay for school uniforms, replace the washing machine, are struggling with bills etc etc, then they should go, or you should massively cut back - for example can you get them done at a college for less money?
However, if you have lots of non essentials in your budget, haven't reduced the cost of your bills by shopping around, spend loads in the supermarket etc etc, then you might be able to free up money elsewhere to keep paying for them if you feel they're worth it.
You can only spend each pound once, so keeping the hair extensions might mean you have to give something else up, so it's up to you to decide.