if possible Please make use of before Save flows (Fast Field Updates), to avoid recursion on Case.
as here we are updating the same record which has triggered the flow.
Once the flow is created , click on + icon shown below the start element and click on "Update triggering Record".
Type Field Name there in Field box, and provide values. i.e. comma seperated emaild Ids. like abcd@xyz.com,efg@hijk.com,pqrs@jklm.com
Even if you are continue using After Save flow only, then update the field with such comma seperated values.

So by doing this your task of populating multiple email Ids in a single field is accomplished.

Now Sending email part, so as we need some logic to perform to determine the to addresses, email logic should be written in trigger code.

Write an afterUpdate trigger on Case, whenver Case status is changed to Closed, invoke email sending method.

in that method get all the email Ids as a String and use "split(',') to convert it to a list of string.

let say cs is your Case record and email__c is the custom long text area field that holds all email Ids.
List<String> allEmailList = cs.email__c.split(',');
now use this allEmailList to set the toAddress while creating instance of Messaging.singleEmailMessage.
For more information on how to send email through apex check this URL:

https://developer.salesforce.com/docs/atlas.en-us.apexcode.meta/apexcode/apex_forcecom_email_outbound.htm
https://developer.salesforce.com/docs/atlas.en-us.apexref.meta/apexref/apex_classes_email_outbound_single.htm

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