Google account without gmail
No, there is no other way. You must have a Gmail account to download from Google Play Store.

From their site:


  To use Google Play to download Android apps, you'll need to have a
  Google account with a Google email (either Gmail or a Google-hosted
  domain) associated with your phone or tablet.


If you don't like Gmail, you don't have to use it. Just disable the syncing for emails, and you'll be fine.