Where does elastic beanstalk t2 server store my node.js app files?

As a more general answer (not node.js-specific), check eb-engine.log from your Elastic Beanstalk environment.

Where are the logs? In Elastic Beanstalk's online interface in the sidebar of your environment.

...
2021/06/12 22:40:34.097454 [INFO] app source bundle is zip file ...
2021/06/12 22:40:34.097460 [INFO] extracting /opt/elasticbeanstalk/deployment/app_source_bundle to /var/app/staging/
2021/06/12 22:40:34.097470 [INFO] Running command /bin/sh -c /usr/bin/unzip -q -o /opt/elasticbeanstalk/deployment/app_source_bundle -d /var/app/staging/
2021/06/12 22:40:34.138333 [INFO] finished extracting /opt/elasticbeanstalk/deployment/app_source_bundle to /var/app/staging/ successfully
...

Interpretation:

1. app_source_bundle was uploaded to /opt/elasticbeanstalk/deployment/ initially
2. app_source_bundle was extracted to /var/app/staging/

If Elastic Beanstalk deployment is successful, you can find the running app's code in /var/app/current (as Manuel pointed out).

It's in the /tmp/deployment/application folder during deployment and the moved to /var/app/current afterward.

In case you search them, the node logs are in /var/log/nodejs/nodejs.log and the application will bind to 8081 no matter what PORT environment variable you set in the Environment Variables in the console.

The top answer seems to be still working in November of 2020 for node app (nuxt app in my case)

/var/app/current