All you need to do is:

go to www.123-reg.com and buy a domain name. I don't quite understand you question though, because you can obviously make up your own domain?

If I have interprated your question wrong please say.

If you mean simply registering / creating your own domain name ...
Sure, its easy - and there are many places to do so.

Some easy and quick ones include:

- http://www.godaddy.com
- http://www.namecheap.com
- http://www.registerfly.com
- http://www.moniker.com
- http://www.google.com/a/ (new one, as mentioned on front page).

... and the list is endless, there are so very many.

When registering a domain just make sure your registrar is ICANN accredited so you know everything is in the right hands.

what is the place to buy cheap co.uk domains??

Well I use www.titanhost.com but I know they are quite expensive at £11 for 2 years but really that isn't alot of money anyway. I think you can get it down to £2-£5 you could go to www.123-reg.com .

I hope that helps you but really there isn't much difference between 5 and 10 pounds...

You can run your own DNS server which should give you the means to call the machines on your own network whatever you want.
But those names will NOT be visible by anyone outside that network.