I'm locked out of my Gmail account as it says it detected unusual activity and asks for my phone number to send a verification code. I don't know why this happened all of a sudden. I don't have a phone number to give them, so can I just wait a day or two and will I be able to log back in like normal?
Most likely you can't wait it out. There is a trick on android if you log in to gmail with the native mail client (or any pop3 basic client) you can bypass things like this on most mail providers.
What you can do is give them a quick phone call and have them sort it out for you pretty quickly. This is a very common problem when it comes to Google based services. People don't have the phone they set up the email with and end up getting locked out. There are plenty of options but Google has amazing live support services that could have it sorted out pretty quick. So long as you can provide the essential account information they ask for.
Remember to always keep your login recovery methods up to date and you will never run in to this problem.