If you have GSuite you can, but I doubt it is possible for individual accounts. [1,2]
You can however send email to google's SMTP server, using an ordinary gmail account [3]. This is how email clients (Eudora, Apple Mail, sendmail) work. The only trick is that they force it to be authenticated and encrypted.
I believe (but have not verified recently) that if you send an email with an email address/domain other than of the authorising account, it will be delivered, even lacking all SPF etc. But, it will block for anything above small quantities of mail, and unless you set up DMARC for your domain, it will still get dropped or marked as spam. [4]
Running an SMTP server for yourself is actually fairly simple (okay, assuming a deep background in linux, but very educational). The real drama is DNS/DMARC/SPF/DKIM and users ruining your reputation.
[1] https://www.datto.com/resource-downloads/SettingUpExternalMa...
[2] https://support.google.com/a/answer/178333?hl=en
[3] https://www.digitalocean.com/community/tutorials/how-to-use-...
[4] https://kinsta.com/knowledgebase/free-smtp-server/