Gmail locked out “detected unusual activity” asking for phone # – will this go away later

There are several options available:

• This form of verification only happens occasionally, so you could try creating another account.
• You could try filling out the support form linked to on that page and select "I don't want to give my telephone number to Google" as the issue. If enough people do this, it may persuade Google to use some other means of verification. I'm not sure if they will activate it for you or not since I went with the last option:
• Choose to verify by voice rather than text message, and enter in your phone number. While this does give them your phone number during the sign-up process, it seems like it isn't stored in your account. You can go to Google Account Settings > Change Password Recovery Options, and you will see that there is only an SMS section, and your phone number isn't stored there. It might still be stored in Google's database behind the scenes though.
• If you are transitioning as part of the Google Apps upgrade process like I am, and your old account had Google Voice, you could enter its phone number, and Google will send the verification to that number.

Update: even if you verify by text message, the phone number won't appear in your settings.

It sounds like someone has been able to "steal" your account and use the security features against you.

You should start with Google Account Recovery and see if you can get your account back