HTTP 503 – Service Unavailable – craft\web\ServiceUnavailableHttpException [installation]
Very likely it cannot connect to the database; that's typically what the 503 error is about:

https://httpstatuses.com/503

...so check the credentials in your .env file, and make sure that the version of MySQL that your website is using is the internal MAMP MySQL, and not the system-wide MySQL (if any).

This video may help you: Installing Craft CMS v3 with MAMP Pro

...and also a discussion of local development environments we had here might be of interest as well: Shootout at the Local Dev Corral