How do I get column datatype in Oracle with PL-SQL with low privileges?

You can use the desc command.

desc MY_TABLE

This will give you the column names, whether null is valid, and the datatype (and length if applicable)

The best solution that I've found for such case is

select column_name, data_type||
case
when data_precision is not null and nvl(data_scale,0)>0 then '('||data_precision||','||data_scale||')'
when data_precision is not null and nvl(data_scale,0)=0 then '('||data_precision||')'
when data_precision is null and data_scale is not null then '(*,'||data_scale||')'
when char_length>0 then '('||char_length|| case char_used 
                                                         when 'B' then ' Byte'
                                                         when 'C' then ' Char'
                                                         else null 
                                           end||')'
end||decode(nullable, 'N', ' NOT NULL')
from user_tab_columns
where table_name = 'TABLE_NAME'
and column_name = 'COLUMN_NAME';

@Aaron Stainback, thank you for correction!

ALL_TAB_COLUMNS should be queryable from PL/SQL. DESC is a SQL*Plus command.

SQL> desc all_tab_columns;
 Name                                      Null?    Type
 ----------------------------------------- -------- ----------------------------
 OWNER                                     NOT NULL VARCHAR2(30)
 TABLE_NAME                                NOT NULL VARCHAR2(30)
 COLUMN_NAME                               NOT NULL VARCHAR2(30)
 DATA_TYPE                                          VARCHAR2(106)
 DATA_TYPE_MOD                                      VARCHAR2(3)
 DATA_TYPE_OWNER                                    VARCHAR2(30)
 DATA_LENGTH                               NOT NULL NUMBER
 DATA_PRECISION                                     NUMBER
 DATA_SCALE                                         NUMBER
 NULLABLE                                           VARCHAR2(1)
 COLUMN_ID                                          NUMBER
 DEFAULT_LENGTH                                     NUMBER
 DATA_DEFAULT                                       LONG
 NUM_DISTINCT                                       NUMBER
 LOW_VALUE                                          RAW(32)
 HIGH_VALUE                                         RAW(32)
 DENSITY                                            NUMBER
 NUM_NULLS                                          NUMBER
 NUM_BUCKETS                                        NUMBER
 LAST_ANALYZED                                      DATE
 SAMPLE_SIZE                                        NUMBER
 CHARACTER_SET_NAME                                 VARCHAR2(44)
 CHAR_COL_DECL_LENGTH                               NUMBER
 GLOBAL_STATS                                       VARCHAR2(3)
 USER_STATS                                         VARCHAR2(3)
 AVG_COL_LEN                                        NUMBER
 CHAR_LENGTH                                        NUMBER
 CHAR_USED                                          VARCHAR2(1)
 V80_FMT_IMAGE                                      VARCHAR2(3)
 DATA_UPGRADED                                      VARCHAR2(3)
 HISTOGRAM                                          VARCHAR2(15)

Note: if you are trying to get this information for tables that are in a different SCHEMA use the all_tab_columns view, we have this problem as our Applications use a different SCHEMA for security purposes.

use the following:

EG:

SELECT
    data_length 
FROM
    all_tab_columns 
WHERE
    upper(table_name) = 'MY_TABLE_NAME' AND upper(column_name) = 'MY_COL_NAME'