How to reinstall messed-up 14.04 without losing personal data

You simply install again and do not format the / (root) partition as you install. If you have a separate home partition, do not format it.

These are options in the advanced partitioning section. This is the "something else" option, you can then manually review partitioning and ensure the root and / or home partitions are not formatted.

See : https://help.ubuntu.com/community/GraphicalInstall

See https://help.ubuntu.com/community/UbuntuReinstallation

Since Hardy it is possible to reinstall Ubuntu without losing the content of the /home folder (the folder that contains program settings, internet bookmarks, emails and all your documents, music, videos and other user files). This can be done even if /home is not on a separate partition (which is the case by default if you did not manually separate it when installing Ubuntu originally).