Way to export Oracle’s UNDO

PostgreSQL does not currently have it. But you are right, with MVCC it should theoretically be possible. In fact, the old university Postgres had an equivalent feature called time travel, but it was later removed because it caused too much overhead and the practical use was limited. While it would probably be possible to resurrect it, since then there have been many optimizations to the MVCC setup that would make it much harder.

Gaius is right, use DBA_TABLES for NUM_ROWS and DBA_SEGMENTS for size:

select owner,table_name,num_rows,last_analyzed from dba_tables;

The num_rows count is as of LAST_ANALYZED date, which should be close enough even without running DBMS_STATS.

For sizing:

select owner,segment_type,sum(bytes)/1024/1024 size_mb
from dba_segments
group by owner,segment_type;

or (depending what level of details you need):

select owner,segment_name,sum(bytes)/1024/1024 size_mb
from dba_segments
group by owner,segment_name;

For LOBs you'll need to join it to DBA_LOBS, for indexes to DBA_INDEXES, for tables to DBA_TABLES. A lot will depend on your specific requirement. Since you mentioned you are trying to figure out used space in tablespace it might be as simple as:

select sum(bytes)/1024/1024 size_mb
from dba_segments
where tablespace_name='YOUR_TBLSP_NAME';

Your solution would really work only with small DB as it would not be feasible to read all data on large DB.

I am hesitant to post how to address ORA-1555 as in this case this it NOT your primary issue but just for completeness - you are on 10g using automatic undo management so your DBA would have to increase undo_retention in your database (the ixora link is relevant to a DB without auto undo management).