Lack of space in SYSTEM and SYSAUX tablespace in oracle 11g R2
The SYSTEM and SYSAUX tablespaces are reuqired by the RDBMS to function correctly. 

The SYSAUX tablespace

A list of objects that reside in the SYSAUX tablespace can be retrieved using the V$SYSAUX_OCCUPANTS view. This view displays the following information:

OCCUPANT_NAME        VARCHAR(64)    Occupant name  
OCCUPANT_DESC        VARCHAR(64)    Occupant description  
SCHEMA_NAME          VARCHAR2(64)   Schema name for the occupant  
MOVE_PROCEDURE       VARCHAR2(64)   Name of the move procedure; null if not applicable  
MOVE_PROCEDURE_DESC  VARCHAR2(64)   Description of the move procedure  
SPACE_USAGE_KBYTES   NUMBER         Current space usage of the occupant (in KB)  


Reference: V$SYSAUX_OCCUPANTS

The SYSAUX table is home for the following components:

SYSAUX Table                         Previous Location
------------------------------------ --------------------------
Analytical Workspace Object Table    SYSTEM
Enterprise Manager Repository        OEM_REPOSITORY
LogMiner                             SYSTEM
Logical Standby                      SYSTEM
OLAP API History Tables              CWMLITE
Oracle Data Mining                   ODM
Oracle Spatial                       SYSTEM
Oracle Streams                       SYSTEM
Oracle Text                          DRSYS
Oracle Ultra Search                  DRSYS
Ora interMedia ORDPLUGINS Comp.      SYSTEM
Ora interMedia ORDSYS Components     SYSTEM
Ora interMedia SI_INFORMTN_SCHEMA C. SYSTEM
Server Manageability Components      -
Statspack Repository                 User-defined
Oracle Scheduler                     - 
Workspace Manager                    SYSTEM


Reference: Database Components and the SYSAUX Tablespace

If you can reduce the size of the data being stored by the components, then you can reduce the size of the required SYSAUX tablespace.

The largest portion of the SYSAUX tablespace is occupied by the Automatic Workload Repository (AWR). The following size recommendations are provided by Oracle to determine the AWR size and to a large portion the size of the SYSAUX tablespace:

Number of CPUs                                 2    |     8    |    32
Number of concurrently active sessions        10    |    20    |   100
Number of user objects: tables and indexes   500    | 5,000    |50,000
Estimated SYSAUX size with default config    500 MB |     2 GB |     5 GB


Reference: Controlling the Size of the SYSAUX Tablespace

If your database has been constantly growing, or if you are monitoring a lot of objects with Enterprise Management Console, then your SYSAUX tablespace will slowly fill up.

The SYSTEM tablespace

The SYSTEM tablespace always contains the data dictionary tables for the entire database. So if your database grows in size due to new objects, then your SYSTEM tablespace will require more space too.

Data Dictionary

What does the data dictionary contain: 


The definitions of all schema objects in the database (tables, views, indexes, clusters, synonyms, sequences, procedures, functions, packages, triggers, and so on)  
How much space has been allocated for, and is currently used by, the schema objects  
Default values for columns  
Integrity constraint information  
The names of Oracle users  
Privileges and roles each user has been granted  
Auditing information, such as who has accessed or updated various schema objects  
Other general database information  


Reference: The Data Dictionary

Solutions:


Reduce the amount of monitored objects in EM
Resize the SYSAUX tablespace according to the recommendations in above table.


ALTER DATABASE DATAFILE '/path/systemSID.dbf' RESIZE 1000M;

Resize the SYSTEM tablespace 


ALTER DATABASE DATAFILE '/path/sysauxSID.dbf' RESIZE 1000M;