I often use 3rd party mailing services (namely mailgun, mandrill but have used others too). You can pay these services a lot to keep all your mail for a while, but the sheer amount of it will always be an issue if you're looking to capture every single mail that goes out.
I have implemented an exim4 configuration that keeps a copy in a local Maildir at times when I've been paranoid and wanted to see exactly what goes out (e.g. check placeholder replacements). This has benefits in that it's on your own server so you can easily do what you want with it: search it, delete it, compress it (or store it on a compressed filesystem, e.g. btrfs offers this option), access it using MUAs or webmail (if you configure it right), store it somewhere affordable - e.g. (Bytemark give you 100GB storage for £2/mo and their cheapest VM is £6/mo).