Neil, we apologize for the frustration that you have encountered.
If you hit any resource quotas on your free tier app service, it should in most cases result in 503 errors, which means your site is stopped and will resume once the resource quota timer resets. If you have not hit any 503 errors, you are likely not encountering any resource quota issues.
The only times I have encountered a 404 error with Eclipse deployed apps were when the customer forgot to transfer the server.xml data sources to context.xml as the default Tomcat server.xml cannot be edited but you're using JSP.
One final item to check would be the diagnose and solve blade of your web app. This is the same powerful tool that our support engineers use to troubleshoot issues. To learn more about it, read about it here. This might help to uncover any issues.
If you have not located the issue and the diagnose and solve blade does not work, please reach out to us at azcommunity@microsoft.com with the subject ATTN: Bryan and in the body of the email include your Azure Subscription ID.
If you have resolve the issue already, we would appreciate it if you shared the solution for the community. We look forward to your reply.