ArcGIS query specific character(s) in specific part of a field

An underscore _ is used in some databases as a single character place wildcard. A percent % is wildcard for all character places.

Therefore, something like "FIELD1" LIKE '_____00%' should work, depending on the type of database. That would use five character wildcards, then your two zeros, then another wildcard for the rest of the string.

The following would select all records that have a zero in the 6th character position as well as in the 7th character position.

FIELDNAME like '_____0%' and FIELDNAME like '______0%'

There are five underscores preceding the zero in the first part of the expression and then six underscores preceding the zero in the second part. The underscore is a 'wildcard' that requires any character to be present (can be a space, a number, a letter, a symbol, etc.). The percent (%) wildcard allows anything to follow (also allows nothing to follow - does not require any character).

A more succinct way to query this would be

FIELDNAME like '_____00%'