Does Linode offer an SMTP relay service?

Linode does not offer an SMTP relay service at this time. However, there are plenty of guides on how to configure an email server on a Linode here.

If you're interested in a SaaS provider and specifically want a relay so you don't have to worry about your own mail server, I can recommend at a personal level that I've used MailGun for transactional emails on their free tier to great success. (<10,000 emails/month)
https://www.mailgun.com/

You may also want to check out:

• https://sendgrid.com/
• https://www.mailjet.com/
• https://www.sparkpost.com/