I’m really sorry that you are facing this dilemma and I can appreciate how difficult it must be for you.
I would also suggest that you speak to Citizens Advice before you decide on what to do. It seems as if you would just have enough to make ends meet so, if you can overcome the transport issue, I would recommend that you take the job so that you have work experience to put on your CV. I’ve never been in this type of situation where you are wondering if a job is worthwhile financially but I would think that the experience would be valuable to you. You never know, the job could lead to better things in 6 months or a year.
Could you try asking people you know if they or anyone they know drives to the same area as the leisure centre? Offering a share of the petrol costs would help both of you. Obviously, this depends upon the working pattern that you have been offered but it is worth thinking about.
I would advise that you pursue every avenue before making a decision. If you can work things out financially, I would encourage you to take the job and get your career going.