AWS recently launched a product for this called Amazon WorkMail. Their pricing page says:
With Amazon WorkMail, there are no upfront fees, no required minimum commitments, and no long-term contracts.
Amazon WorkMail costs $4 per user per month and includes 50 GB of mailbox storage for each user. You can get started with a 30-day free trial for up to 25 users.
$4 per month is competitive email hosting pricing. It isn't the cheapest, but it is cheaper than some of the big players in the space such as hosted Gmail (Google G Suite.)
Keep in mind that you are free to choose any hosted email service that you want. Even if you control the domain entirely through AWS, you can set the MX records for mail to point wherever you want. If you are looking to compare providers search Google for things like "hosted email reviews" or "email mailboxes reviews."