First question - 'Is gmail safe for online accounts?'
Yes, with caveats. Google takes its role as federated identify management seriously and works incredibly hard in the background to keep their services safe. You can read more about their security philosophy here: https://www.google.com/about/appsecurity/
The caveat is that the user tends to be the weakest link is email account. For example, if you maintain poor security hygiene, Google's security won't mean much. For example, if you do one of the following, it doesn't matter if you use encrypted email, protonmail, gmail, or your ISP:
- Click on a phishing link and type in your username and password
- Use the same username and password on multiple sites
Phishing comment - If you click on a phishing link and give up your username and password, an attacker can break into your email account and do whatever nefarious business they want to do. This attack can be mitigated if you use 2 Factor Authentication.
Multiple Site comment - Whenever a site, such as a web forum, gets hacked and credentials get stolen, many criminals will look at the collection of usernames and passwords (usernames tend to be email addresses) and go to the email provider and try out the username and password. If you use the same password on your email account as you do throughout the web, then if those other sites get popped your email account is at risk. This can be mitigated by following what Serge wrote above about using a password manager.
Second - 'Hacking gmail in 30 minutes'. I doubt he has 0-days at his disposal to use on Google, though if he does Zerodium would pay him handsomely. More likely he did a lateral attack, like sniffing unencrypted traffic or phishing his friends. (Or, like the mob, taking a wrench to someone's knees for the un/pw). It's an unlikely comment.
Third - 'What email services?' - This is asking the wrong question. Google, for now, is a good company with a lot of talented security professionals working to keep things secure. Are they the best? I don't know how to judge 'best', but they are good. This is like asking what Toothpaste to use to keep your teeth clean? It's not about the toothpaste, it's about the brush and your hygiene. Likewise, it's not about gmail vs yahoo vs your ISP, it's about your personal security hygiene (as mentioned above)
Hope that helps.