If you have a
sendmail-compatible MTA, crond will use it to send mail, like most other Unix programs do.
If you want to forward everything to an "outside" mailbox, you need to install a MTA. For a server, I prefer
postfix - but there also are
exim4 and the old
sendmail, after which that binary was named.
For a personal computer, where you don't need incoming mail, you can get
esmtp - it relays mail through an external SMTP server, such as Gmail's or your ISP's. (You probably will need to get a MDA such as
maildrop and configure esmtp for it. It's still more lightweight than having five
postfix daemons running.)
Then, create a file
~/.forward with your real email address in it, and everything should be forwarded there.
/var/usr/emails/$USER is quite an unusual place to store mail... for me, at least.