Oracle stores its data in tablespaces which have datafiles assigned to them that provide the actual space. So a couple of questions:
1. Are you out of space on the entire server the database resides on?
If so, you need to add disks to the server, then add a data file to your tablespace that resides on the new disk.
if not, you can add a new datafile on an existing disk to your tablespace. Or you can modify the existing datafiles and turn autoextend on. That will continue to increase the size of your datafiles until you hit the Oracle Max size (depends on blocksize, 16K is 64GB).
2. Are your tables in a separate tablespace than SYSTEM?
If so, MOVE THEM! filling up the system tablespace will crash your database.