Why did the IT Crowd choose to use a real telephone number?

It can't exist as a real phone number, with or without the Reading code, even if you just take the first 11 digits.
You just can't have 999 as a dialing code - for what I would have thought would be fairly obvious reasons.

The joke was based on the then recent transition from the previously simple 192 to get directory enquiries, to a plethora of private enterprises - all of which started with 118, the most famous being 118 118. So they took the closest they could get to 118, which just happens to be Reading's 0118, then added a whole slew of unmemorable numbers behind it, borrowing 999 too.

However, though few people will be able to test it these days, someone at Google thought that it would be amusing to add it as an Easter Egg in Android phones.
If you dial that number into an Android Marshmallow [OS 6] phone, the dialling button will flash red & blue, emergency-services-like.

A gif of that & the full backstory is available on Android dialer: IT Crowd's 0118 999 881 999 119 725 3 easter egg.

The situation is simple, all number ranges in the UK are controlled by OFCOM (Oftel as was). Each operator has to apply for a range of numbers in each area. OFCOM will not issue certain numbers if they will cause a conflict, such as local code starting with 999 etc. The 118 range has 118999 set to protected so no UK operator will be able to use that as a subscriber number. The extract from the relevant document for 118999 shows:

1189 88 Allocated BT 3+7 08/04/1996

1189 89 Allocated BT 3+7 08/04/1996

1189 90 Allocated Verizon UK Limited 3+7 27/02/2006

1189 91 Allocated Colt Technology Services 3+7 11/05/2006

1189 92 Protected 3+7

1189 93 Allocated Nextec UK Limited 3+7 06/12/2013

1189 94 Quarantined 3+7 26/09/2006

1189 95 Allocated Simwood eSMS Limited 3+7 18/07/2006

1189 96 Allocated TalkTalk Communications Limited 3+7 15/06/2006

1189 97 Allocated Spitfire Network Services Limited 3+7 27/06/2006

1189 98 Allocated Served Up Limited 3+7 09/05/2008

1189 99 Protected 3+7

1190 Protected 4+6

1191 Protected 4+6

All the files are (currently) found at https://www.ofcom.org.uk/phones-telecoms-and-internet/information-for-industry/numbering/numbering-data

With Number Portability (oh what fun sorting that out was) the actual operator with a particular telephone number is different.

Edit re Quarantined ; It can mean that the number range has been allocated but the recipient operator has ceased to exist or for some other reason it is not using the number range. So it is put into isolation for a period of time until it can be safely re-allocated. Also, the TV etc numbers for Reading are:Reading 0118 496 0000 to 496 0999

I used to work with telephone exchanges in the UK in about that era. Once upon a time there were national and district dialling codes and a short phone number, but by this point the district code was part of the local phone number. But they kept the old short numbers that did emergency, directory enquiries etc.

You could either dial the national number, or if you were in the same area, just the local number. Any number beginning with one of the short codes was not used, because it would connect you as soon as it recognised those digits. So if you tried to dial 999123 you would always get the emergency number. But 199923 was available to use, and also 123999 because the exchanges no longer recognised the old district-local dialling.

Numbers beginning with 0800 couldn't be dialled without the 0800. So they could allow 0800111999 or 0800999111 if they chose. I'm not sure if they did.

I also understand that they reserved 911 and other emergency numbers, which also routed to the same emergency line. Basically, if you're overseas and in an emergency, they can't expect you to read the instructions first!

EDIT: Sorry, 123 is the speaking clock. Forgot that. I just used it to represent any unallocated number.