Should I bother trying for Oxford

I am sorry to hear of your ill health and hope that you are recovered.
I can't see that taking a fourth A level would add much to your chances, as Oxford only wants three A levels for Classics, although combined with a re-sit it might show commitment, I suppose.
You (preferably assisted by your school) could try to persuade Oxford via personal statement and reference that your medical circumstances were such that your AAB should be treated as though it were AAA, but that may be difficult.
What subject is the B in? If it's a classical language, ancient history, or philosophy, that may count against you.
Otherwise, try a resit, or accept that things are as they are, and go to another good university. There are several places where you could study classics and have a good time. Not getting into Oxford won't determine your life chances or happiness. Oxford may be available to you for postgrad if you get a first and fancy some further study.

Thank you for your response! The B was in English Literature.

Then you should apply to Oxford University!!! 😀
You could take A-Level Maths perhaps? Because the YouTube channels:
BicenMaths, TLMaths, ExamSolutions, and HindsMaths got almost everyone either an A or A* grade.

I’m not sure this is great advice, and I think it’s unlikely Oxford would care about Maths when OP is hoping for a Classics degree. In circumstances where the fourth A-Level is irrelevant (let’s imagine someone is doing English, Latin, Ancient History and then Biology), Oxbridge may ignore it or specify that your grades should be in only the relevant subjects.
OP, do you have Latin and Greek A-Level? If you don’t, you could consider resitting your B subject and also doing one of those.

I’m not sure this is great advice, and I think it’s unlikely Oxford would care about Maths when OP is hoping for a Classics degree. In circumstances where the fourth A-Level is irrelevant (let’s imagine someone is doing English, Latin, Ancient History and then Biology), Oxbridge may ignore it or specify that your grades should be in only the relevant subjects.
OP, do you have Latin and Greek A-Level? If you don’t, you could consider resitting your B subject and also doing one of those.

How would I go about doing Latin or Greek A-level? I have Latin at GCSE but because of my illness I missed year 11 teaching and didn’t perform too well.

A poor or middling GCSE result in Latin may be another obstacle to a competitive application to Oxford to read Classics, unless your mitigating circumstances are very strong. You could in theory have a go at A level Greek ab initio, but that might not be a good idea. A level Latin would make more sense.

I honestly think Greek is much harder than Latin and so it probably isn't feasible to do A-Level Greek ab initio. Your suggestion about Ancient History actually might be the best idea — I’ve just realised that if OP did Latin, they would probably need to do the Latin CAT (in essence, a translation test). The CAT isn’t impossibly hard, but it is difficult, and it’s also in October. If OP is starting from (likely rusty) GCSE Latin, then finding a tutor and learning the necessary grammar and vocab required to do well in the CAT in just over a month might not be plausible. Of course, it’s possible that OP won’t need to do the CAT if they were starting Latin A-Level now — this is probably something to ask their chosen college about.