I signed up to Demon internet last year, which was costing me around £20 a month.

I
cancelled the contract a few weeks back (with 1 month notice), as I am
moving to a different house and do not want to continue the service.

However
- I received a phone call yesterday from their customer services team,
letting me know that no direct debit payments had been taken since I
joined, and that I now owed £214.74!

I had not noticed that
this was the case - and he could not give me any explanation about why
this happened - he simply said "it is one of those things!"

 Do I
have any option other than paying the money to them? I am extremely
annoyed that 1) nobody noticed the payments were not being taken, and
2) that they cannot provide me with a satisfactory explanation!

Anyone else had experiences like this? What should I do next?

If you signed a contract and obtained and used the service, then you are legally obliged to pay for it.

Again, why are you so angry? Do you not check your accounts? Why is this sort of thing always "... someone else's fault ..."? Yes, it is inconvenient and yes it should have been spotted, and now you are angry because someone spotted the mistake and are asking - legitimately - for their money.

You ask what you should do next - that indicates everything this society has become. You should pay them what you owe. What else is there to do?

SHARK!