Can numbers be illegal?
This video explaining this wikipedia article says that possessing a particular number can be illegal. Is this true?
If your question, as given in the topic, is interpreted literally, without the context from that video and article and without any possible other interpretations, then the answer is clearly and absolutely "no".
It is not possible that "possessing" a particular number can be illegal, in any form or fashion.
All cases mentioned in the other answers, as well as in your text and video, are not the possession of a "number", but of some information "xyz" which happens to contain a number as part of it.
Let me give you a simple example: say I write a diploma thesis and count the number of cars that drive by my home during a day. I end up at 322884 (it's a busy street). It just so happens that the PIN for launching the nuclear arsenal of my country also happens to be 322884. If it were illegal to possess that number, I would go to jail.
Sure, I do "possess" that number. It is clearly written on my hard drive, I have printed it on my thesis, I have even publicised it. So if the NSA does a full-out scan of all my property, they would easily find it. I did not possess the information that it is the nuke-PIN, though. So I will most certainly not go to jail for it.
TL;DR: yes, it can be illegal to possess a certain information. No, it cannot be illegal to possess a certain number (without connotation).