Can I make a full export of a corrupted database?
By default the full scan on that table with a corrupt block will fail, but you can make the scans skip the corrupt blocks of an object:

DBMS_REPAIR.SKIP_CORRUPT_BLOCKS


  This procedure enables or disables the skipping of corrupt blocks
  during index and table scans of the specified object.


DBMS_REPAIR.SKIP_CORRUPT_BLOCKS (
   schema_name  IN VARCHAR2,
   object_name  IN VARCHAR2,
   object_type  IN BINARY_INTEGER DEFAULT TABLE_OBJECT,
   flags        IN BINARY_INTEGER DEFAULT SKIP_FLAG);


In RMAN you can create a backup with corrupt blocks by specifying the allowed maximum amount of corrupt blocks:

SET


  MAXCORRUPT FOR DATAFILE datafileSpec TO integer
  
  Sets a limit on the number of previously undetected block corruptions
  that the database permits in a specified data file or group of data
  files. The default limit is zero, meaning that RMAN tolerates no
  corrupt blocks.
  
  The SET MAXCORRUPT command specifies the total number of physical and
  logical corruptions permitted in a data file during a backup job. If
  the sum of physical and logical corruptions detected for a data file
  is no more than its MAXCORRUPT setting, then the BACKUP command
  completes. If more than MAXCORRUPT corrupt blocks exist, then RMAN
  terminates without creating output files.


But you should just fix the object with the corrupt block instead of ignoring it.