I'm not going to tell you outright not to do it. It sounds fun, and SF is a good location to be in tech right now. However, there are some major red flags around you quitting your job, moving to one of the most expensive locales in the world, for an unpaid role.
* Be prepared for a potentially lengthy period of employment. The market is not what it once was, especially early in your career, and interview cycles are long.
* Consider that startup founders in SF have a high appetite for risk, may be very charismatic and persuasive, and ultimately will bear none of the consequences if this doesn't work out well for you. Sadly, founders can be exploitative too, and asking you to move to SF to work for free sounds exploitative.
* There are a lot of good options out there! Speaking as someone who broke into the SF tech scene after starting my career outside of it, the first few companies/job offers are going to sound unbelievably amazing. That's because they are sales pitches, sold to you by professional salespeople. Over time, once you have seen some of those promises fail to work out, you may view the sales pitches with skepticism. There are going to be more amazing opportunities, and some of them will pay.
Having said all that: I once quit my job and spent down my savings to work on a passion project. And I once left my hometown and set sail for SF, too (although I was paid for that.) Ultimately, if you have a good social safety net and good skills, you will probably be alright, even if you do something a little bit wild like this.