How many extracurriculars should I take if I want to get into Yale?

The number doesn't matter, usually most colleges only allow you to add up to five extracurriculars. It's not the number, what matters is what you DID during your time in a club. If you want to boost chances via ECs, you need to do something impressive, e.g. be a president/founder of a service club at school at raised $100,000, being the presider/founder of a music club that won a national music/choir competition, etc. Or at least this is what you should be doing by the end of junior/early senior year to impress top colleges with ECs.

The question is never how many, but the quality/impact of your work within each activity. If you have 200 hours of community service, is it better to have 20 each for 10 clubs or 100 between 2 clubs? This is also why it's best to join extracurriculars with a growth mindset: unless you're just doing it for fun, you want to advance towards some sort of leadership position or achievement. If you're already a rising junior building your college list or a senior applying, it might be a little late to build your resume much unless you get these leadership roles or some sort of award/title soon.

Keep in mind that Yale is one of the most selective schools in this country, and unless you have done 1-2 activities with a lot of impact on your community or some equivalent achievement, with grades to match, your odds are very slim. If you/your family run the net price calculator for Yale, and you can afford the estimated cost you'll get, possibly with loans, I'd suggest EDing.