Yes, ideally you need professional liability insurance in case anything goes wrong and a client needs to claim money from you for damages.
For instance, if they suddenly have a bad reaction to the hair colour and decide to sue you.
Also, supposing their toddler grabbed your hot straighteners and burnt himself and they decide to sue you?
What if you accidentally spilt colour/bleach on the client's carpet?
What if your hairdryer develops a fault and fuses their electrics?
I think it's a higher risk for you when you're working in the client's home as you have no control over your workspace so you might be having to work around children and other family members.
Can you afford to cover your own legal fees and pay compensation?
Salon Gold is often recommended here but just google hairdressing insurance for a selection of providers and try to get at least 3 quotes so that you can compare the cost and the cover provided. Some companies will offer to cover your stock and equipment too.
You're not legally obliged to purchase insurance but it's not that expensive for peace of mind and being properly insured and registered as self employed demonstrates to potential clients that you are a competent professional hairdresser and not an amateur diy-er.