G Suite Admin can not directly access users emails, however, Yes, he has following options to look at any users emails-:

(i) Google Vault ( Read My Complete Google Vault Guide here

G Suite has different plans, and one of them is called “G Suite for Business” which comes an application called “Google Vault“, which saves a copy of all users emails, on the record chat, group messages, files in Google Drive and Team Drive.

So even if a user deletes his/her email or a file in Google Drive, you can login to Google Vault as an Admin and search for users emails.

Please note -: above vault based solution to access your users email will not work with G Suite basic plan as Google does not vault with it, however it can be purchased additionally as shown in Google Vault Pricing guide here.

(ii) Email Delegation -:

G Suite offers an email delegation feature where you a user can delegate his or her gmail mailbox to someone (e.g a CXO delegating to executive assistant), this is usually done by a user, however G Suite Admin can also do it via Gmail API without even users noticing it.

Note-: Though G Suite Admins can setup email delegation behind the scenes, if you are a user you can go to your delegation settings (Gmail –> Settings –> Accounts –> Grant Access to your account) and check if their account is delegated to someone and can delete the delegation too.

(iii) G Suite Content Compliance Rule -:

G Suite Admin can also setup a rule in admin console to trigger a bcc copy of all (or required) users email as shown in the video tutorial above, and this solution works with all G Suite paid plans.

(iv) G Suite Admin Audit API -:

In case if your requirement is not fulfilled by above solutions, you may consider putting a custom solution based on your needs with G Suite Email Audit API. You also don’t need to start from scratch here, if you know a bit of Google Apps Script, you can use this OAuth 2 library to easily use Audit API within Apps Script.