Most email apps only keep downloaded messages on your device for a few days. They do stay on your mail providers servers until you delete them. Most email providers have a web interface or an app to access your email. Go to the webpage of your email provider (hotmail, gmail, yahoo, xfinity, etc) and log into your email account and/or download their app. You will then be able to access all your emails.
Do you have a laptop/desktop that has Outlook? Add your account(s) to Outlook and you should be able to access all emails from there.
I personally use Gmail app and it has never removed emails, they stay there until I delete them or archive them. If it has I am not aware of it. I mainly use my laptop to check emails when I am home. I just make sure that each each of my email address are synced so if I delete or archive them on my laptop it also does it on my phone as well and vice versa.
