The log said:

2014-10-07 10:38:41 1XbVuT-0005Wl-W6 <= www-data@mysite.com U=www-data P=local S=451
2014-10-07 10:38:42 1XbVuU-0005Wq-0n <= www-data@mysite.com U=www-data P=local S=447
2014-10-07 10:38:42 1XbVuU-0005Wq-0n ** contato@mysite.com: Unrouteable address
2014-10-07 10:38:42 1XbVuU-0005Wv-1a <= naoresponda@mysite.com U=www-data P=local S=677 id=5433faf2077ea@mysite.com
2014-10-07 10:38:42 1XbVuU-0005Ww-1Z <= <> R=1XbVuU-0005Wq-0n U=Debian-exim P=local S=1266
2014-10-07 10:38:42 1XbVuU-0005Wq-0n Completed
2014-10-07 10:38:42 1XbVuU-0005Wv-1a ** suporte@mysite.com: Unrouteable address
2014-10-07 10:38:42 1XbVuU-0005Ww-1Z == www-data@mysite.com R=local_user T=maildir_home defer (13): Permission denied: cannot chdir to /var/www/Maildir
2014-10-07 10:38:42 1XbVuU-0005X3-2q <= <> R=1XbVuU-0005Wv-1a U=Debian-exim P=local S=1505
2014-10-07 10:38:42 1XbVuU-0005Wv-1a Completed
2014-10-07 10:38:42 1XbVuU-0005X3-2q ** naoresponda@mysite.com: Unrouteable address
2014-10-07 10:38:42 1XbVuU-0005X3-2q Frozen (delivery error message)

Were you able to solve your problem? Are you running the latest version of exim?

The error persists.

I downloaded the latest version.

Can you post the contents of /etc/aliases?

The content o etc/aliases is:

postmaster: root
root: digitalocean

The erro log says to:

2014-10-23 14:45:17 2XhLVl-1237nQ-Mg <= root@my-site.com U=root P=local S=2207
2014-10-23 14:45:17 2XhLVl-1237nQ-Mg ** digitalocean@my-site.com root@my-site.com: Unrouteable address
2014-10-23 14:45:17 1XhLVl-0007nU-NQ <= <> R=2XhLVl-1237nQ-Mg U=Debian-exim P=local S=3064
2014-10-23 14:45:17 1XhLVl-0007nU-NQ ** digitalocean@my-site.com root@my-site.com: Unrouteable address
2014-10-23 14:45:17 1XhLVl-0007nU-NQ Frozen (delivery error message)
2014-10-23 14:45:17 2XhLVl-1237nQ-Mg Completed

digitalocean@your-site.com does not exist, you need to either create a user for it (How To Add and Delete Users on an Ubuntu 14.04 VPS | DigitalOcean) or set it as an alias to an existing user:

/etc/aliases:

postmaster: root
root: existing user name
contact: existing user name
support: existing user name

and run

sudo newaliases
sudo service exim restart

Try removing root: root and running:

sudo newaliases
sudo service exim restart

again. Does that fix it?

Nope, dont work.

The error log say:

2014-10-23 16:59:23 1XhNbW-0004TC-VT == /var/mail/mail (root@my-site.com) contato@my-site.com R=mail4root T=addressfile defer (13): Permission denied: creating lock file hitching post /var/mail/mail.lock.my-site.com.5449500b.00004326 (euid=8 egid=8)
2014-10-23 16:59:24 1XhNbW-0004TC-VT => personalemail@gmail.com R=dnslookup T=remotesmtp H=gmail-smtp-in.l.google.com [74.125.28.27] X=TLS1.2:RSAARCFOURSHA1:128 DN=“C=US,ST=California,L=Mountain View,O=Google Inc,CN=mx.google.com” C=“250 2.0.0 OK 1414090764 uh7si2269655pac.156 - gsmtp”
2014-10-23 16:59:24 1XhNbY-0004TK-FY <= <> R=1XhNbW-0004TC-VT U=Debian-exim P=local S=6692
2014-10-23 16:59:24 1XhNbY-0004TK-FY ** naoresponda@my-site.com: Unrouteable address
2014-10-23 16:59:24 1XhNbY-0004TK-FY Frozen (delivery error message)

Most important:

Permission denied: creating lock file hitching post /var/mail/mail.lock.my-site.com.5449500b.00004326