Database failures (or the possibility of such) must be assumed to be lurking, ready to strike at some time in the future. A prudent database administrator will plan for the inevitability of failures of all possible kinds, and will have appropriate plans and procedures in place before the failure occurs.
Database recovery is necessary in the event of hardware or software failure. There are several categories of failures; some of these require relatively minor adjustments to the database, while others may depend on the existence of previously prepared database dumps and other recovery data sets. It should be emphasized that if your data is important and/or difficult to regenerate, then you should have considered and prepared for various failure scenarios.
A filled data disk may result in subsequent corruption of database indexes, but not of the fundamental data tables. 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 will refuse to start up. You must free up additional space on the disk (or move the WAL area to another disk; see Section 11.3) and then restart the postmaster to recover from this condition.