Firstly, redundancy is never guaranteed, I've seen people who thought it was almost a "right" and when their company got bought out and the new owners changed T&C's, they got left with statutory and that was it.
Secondly, and again I've seen this happen, depending on immediate family on money matters can end in tears.
To me, 5k would be too low, way too low but your emergency fund doesn't need to be in an instant access bank account, as long as you can get it reasonably quickly and have a credit card (for example) to tide you over, it may suffice.
in a nutshell, if the house burnt down tomorrow or something happened not covered under PHI/leifcover and it was going to take a few months for an insurance payment and you had to cover a mortgage and rent for a year, could you do so?