After years of thinking that all I wanted was a PhD, I finally got a place at a good university: both the advisor and the project sound great, but on the other hand I hate the city where I'd have to live and I am beginning to have doubts about wanting to stay in academia. Plus, I'm not getting any funds, meaning I have to pay the university fee, which is significant, plus support myself, all on my own. I accepted it a few months back but now that I am about to start, I'm not sure this is the best for me. Everyone reassured me that it is much easier to get funding in the second year but I don't know if I can believe that: in case I didn't get it, I would have to work part-time, meaning it would just take forever to get this degree done. Because I've wanted it for so long and because the uni has a good reputation I feel like I should at least give it a go, but I wonder, is it even worth it if I have zero funding? I'm in the humanities so finding jobs ain't easy, I just don't know if I'm making it harder for myself right now.

As someone doing a humanities PhD right now, I wrestled a lot for the self-funding aspect (only one research council funds my PhD area, and I was rejected). I would seriously warn against starting a PhD with the mindset that you will definitely get funding in Years 2 onwards, because that very well might not happen. I have heard of many people feeling trapped in their PhD because they started it thinking that they would get funding later, but it never happened, but they had already sunk a whole year into the PhD by that time, so felt like they couldn't quit. I would also say that after mental health, financial issues is the reason many PhD students drop out or don't complete.
I'd say that if you are really passionate about your project, that might be able to motivate you through; are there places close to the city that you could live in instead and commute in?

Thank you for your help. I do like my project and am passionate about my subject in general, and I guess I could find accommodation in a prettier city nearby...it's just that I realise how I might feel trapped as you described, and because of lack of funding, the city and everything I am not going in particularly motivated. Idk, it feels very demotivating and worrying to know that I might never get funding and I might have to quit because of that, plus I have this feeling that this isn't right for me, but maybe everyone feels a little bit like that before starting. Sorry you had to struggle with finding funding too!

Also, do you/does anyone know if it is possible to start my unfunded PhD and look for other funded options in the meantime? I mean, would I be considered at all by a new university if I have already started somewhere else? And, in case it is possible, would I be allowed to transfer my research or would I need to present a new project?

You can definitely look for funding during your PhD, but it's unlikely to be huge amounts like studentships are. You can look for grants and scholarships offered by relevant organisations and charities, but obviously don't apply for ones for new PhD students if you've already started yours.

As to whether you'd be considered elsewhere - that's tricky. If you aren't funded, you aren't tied to your supervisor/university in that respect, but your prospective universities may be concerned about why you want to transfer. Whether you can move your research over may be down to your supervisor and how far you've got with your project. A PhD student who started before me had to leave just after her data collection had finished, but the university kept the data and the 'idea' for the project and assigned it to someone else. Equally, a new supervisor may not be happy to get involved with a project which they've had no input on for years.

If you aren't happy with your current PhD offer, and you aren't funded. it might be better to wait and apply to other places.