It sounds like LR is wanting to write some temporary files to do system-defined $TMPDIR or $TMPD and cannot due to either a permissions problem or folders that don’t exist, anymore, and LR cannot recreate them.
Here is a similar discussion, rather long, about the same issue, where it doesn’t end with the user saying things are ok, but maybe what the help said did eventually work:
http://www.justanswer.com/software/7m3ey-trying-open-lr-box-comes-saying-lr-cannot-start.h tml
