When you sign up for Gmail -- or any Google Account -- the registration form asks you to provide a mobile phone number, but you don't need to fill this field. Skip over it and you can submit the form without giving a phone number. If you don't solve the CAPTCHA, however, Google will still ask for a number to verify your account.
The Google registration page displays a CAPTCHA, an image of numbers or letters that you type to prove you're a real person. The page lets you Skip This Verification, but If you do, Google requires phone verification. If you can't read the image, click the circular Get a New Challenge icon rather than skipping the CAPTCHA.
Phone Verification Workarounds
Google offers two options for phone verification: a voice call or a text message. Even though the registration page asks for a mobile phone number, you can provide a landline number and use the voice call option to complete verification. You could also use a friend's phone temporarily and remove the number from your account after verification.
Phone Numbers as a Recovery Method
Even if you don't need to verify your account, having a phone number tied to your Google Account gives you a method to recover your account if you lose your password. As an alternative recovery method, enter another email address you use on the signup page.