The most important disk monitoring task of a database administrator is to make sure the disk doesn't grow full. A filled data disk may result in subsequent corruption of database indexes, but not of the tables themselves. If the WAL files are on the same disk (as is the case for a default configuration) then a filled disk during database initialization may result in corrupted or incomplete WAL files. This failure condition is detected and the database server will refuse to start up.
If you cannot free up additional space on the disk by deleting other things you can move some of the database files to other file systems and create a symlink from the original location. But note that pg_dump cannot save the location layout information of such a setup; a restore would put everything back in one place. To avoid running out of disk space, you can place the WAL files or individual databases in other locations while creating them. See the initdb documentation and Section 18.5 for more information about that.
Tip: Some file systems perform badly when they are almost full, so do not wait until the disk is full to take action.