apt update error: The repository 'http://us.archive.ubuntu.com/ubuntu saucy Release' does not have a Release file

This happens because you still have somehow the sources.list file for Ubuntu 13.10 on your machine:

You can use sudo -i in terminal and copy & paste the following (complete block) into it:

cat > /etc/apt/sources.list << EOF
deb http://archive.ubuntu.com/ubuntu xenial main restricted universe multiverse
deb-src http://archive.ubuntu.com/ubuntu xenial main restricted universe multiverse
deb http://archive.ubuntu.com/ubuntu xenial-updates main restricted universe multiverse
deb-src http://archive.ubuntu.com/ubuntu xenial-updates main restricted universe multiverse
deb http://archive.ubuntu.com/ubuntu xenial-backports main restricted universe multiverse
deb-src http://archive.ubuntu.com/ubuntu xenial-backports main restricted universe multiverse
deb http://archive.ubuntu.com/ubuntu xenial-security main restricted universe multiverse
deb-src http://archive.ubuntu.com/ubuntu xenial-security main restricted universe multiverse
#deb http://archive.ubuntu.com/ubuntu xenial-proposed restricted main universe multiverse
#deb-src http://archive.ubuntu.com/ubuntu xenial-proposed restricted main universe multiverse
deb http://archive.canonical.com/ubuntu xenial partner
deb-src http://archive.canonical.com/ubuntu xenial partner
EOF

This will overwrite your /etc/apt/sources.list file. After that you should use exit to get out of root.

If you not wish all package sources to be enabled you can change this from the GUI applet called 'Software&Updates' which you find within your settings.

Then you should do the obligatory:

sudo apt clean
sudo apt update
sudo apt dist-upgrade

Hope, I'm not late to the party

For anyone that might still face the same issue, the solution above did not work for me. In my case, the root directory partition was full (100% used up), I used

df -h
cd /
du -sch * --exclude=home

to check what was using up the disk space. In my case, there was a backup of mysql data folder in the root directory taking up about 3.7GB out of the 9.1GB space. All I did was to deleted the mysql folder. But be careful of what you delete.

Get sources.list from Ubuntu Sources List Generator and replace your /etc/apt/sources.list with the downloaded file. Next run

sudo apt update
sudo apt dist-upgrade