Oracle - need to extract text between given strings

I'm not that familiar with the POSIX [[:print:]] character class but I got your query functioning using the wildcard .. You need to specify the n match parameter in REGEXP_REPLACE() so that . can match the newline character:

WITH phrases AS (
  SELECT 'stackoverflow is awesome. Begin beginHello,
 World!End end It has everything!' AS phrase
    FROM dual
)
SELECT REGEXP_REPLACE(phrase, '.+Begin begin(.+)End end.+', '\1', 1, 1, 'n')
  FROM phrases;

I used the \1 backreference as I didn't see the need to capture the other groups from the regular expression. It might also be a good idea to use the * quantifier (instead of +) in case there is nothing preceding or following the delimiters. If you want to capture all of the groups then you can use the following:

WITH phrases AS (
  SELECT 'stackoverflow is awesome. Begin beginHello,
 World!End end It has everything!' AS phrase
    FROM dual
)
SELECT REGEXP_REPLACE(phrase, '(.+Begin begin)(.+)(End end.+)', '\2', 1, 1, 'n')
  FROM phrases;

UPDATE - FYI, I tested with [[:print:]] and it doesn't work. This is not surprising since [[:print:]] is supposed to match printable characters. It doesn't match anything with an ASCII value below 32 (a space). You need to use ..

UPDATE #2 - per update to question - I don't think a regex will work the way you want it to. Adding the lazy quantifier to (.+) has no effect and Oracle regular expressions don't have lookahead. There are a couple of things you might do, one is to use INSTR() and SUBSTR():

WITH phrases AS (
  SELECT 'stackoverflow is awesome. Begin beginHello,
 World!End end It has everything!End endTESTTESTTEST' AS phrase
    FROM dual
)
SELECT SUBSTR(phrase, str_start, str_end - str_start) FROM (
    SELECT INSTR(phrase, 'Begin begin') + LENGTH('Begin begin') AS str_start
         , INSTR(phrase, 'End end') AS str_end, phrase
      FROM phrases
);

Another is to combine INSTR() and SUBSTR() with a regular expression:

WITH phrases AS (
  SELECT 'stackoverflow is awesome. Begin beginHello,
 World!End end It has everything!End endTESTTESTTEST' AS phrase
    FROM dual
)
SELECT REGEXP_REPLACE(SUBSTR(phrase, 1, INSTR(phrase, 'End end') + LENGTH('End end')), '.+Begin begin(.+)End end.+', '\1', 1, 1, 'n')
  FROM phrases;

Try this regex:

([[:print:]]+Begin begin)(.+?)(End end[[:print:]]+)

Sample usage:

SELECT regexp_replace(
         phrase ,
         '([[:print:]]+Begin begin)(.+?)(End end[[:print:]]+)',
         '\2',
         1,  -- Start at the beginning of the phrase
         0,  -- Replace ALL occurences
         'n' -- Let dot meta character matches new line character
)
FROM
  (SELECT 'stackoverflow is awesome. Begin beginHello, '
    || chr(10)
    || ' World!End end It has everything!' AS phrase
  FROM DUAL
  )

The dot meta character (.) matches any character in the database character set and the new line character. However, when regexp_replace is called, the match_parameter must contain n switch for dot matches new lines.

In order to get your second option to work you need to add [[:space:][:print:]]* as follows:

with phrases as (
  select 'stackoverflow is awesome. Begin beginHello, 
  World!End end It has everything!' as phrase
    from dual
         )
select regexp_replace(phrase
     , '([[:print:]]+Begin begin)([[:print:]]+[[:space:][:print:]]*)(End end[[:print:]]+)', '\2')
  from phrases
       ;    

But still it will break if you have more \n, for instance it won't work for

with phrases as (
  select 'stackoverflow is awesome. Begin beginHello, 
  World!End end 
  It has everything!' as phrase
    from dual
         )
select regexp_replace(phrase
     , '([[:print:]]+Begin begin)([[:print:]]+[[:space:][:print:]]*)(End end[[:print:]]+)', '\2')
  from phrases
       ;       

Then you need to add

with phrases as (
  select 'stackoverflow is awesome. Begin beginHello, 
  World!End end 
  It has everything!' as phrase
    from dual
         )
select regexp_replace(phrase
     , '([[:print:]]+Begin begin)([[:print:]]+[[:space:][:print:]]*)(End end[[:print:]]+[[:space:][:print:]]*)', '\2')
  from phrases
       ;   

The problem of regex is that you might have to scope the variations and create a rule that match all of them. If something falls out of your scope, you'll have to visit the regex and add the new exception.

You can find extra info here.

 Description.........: This is a function similar to the one that was available from PRIME Computers
                       back in the late 80/90's.  This function will parse out a segment of a string
                       based on a supplied delimiter.  The delimiters can be anything.
Usage:
     Field(i_string     =>'This.is.a.cool.function'
          ,i_deliiter   => '.'
          ,i_start_pos  => 2
          ,i_occurrence => 2)

     Return value = is.a

FUNCTION field(i_string           VARCHAR2
              ,i_delimiter        VARCHAR2
              ,i_occurance        NUMBER DEFAULT 1
              ,i_return_instances NUMBER DEFAULT 1) RETURN VARCHAR2 IS
  --
  v_delimiter      VARCHAR2(1);
  n_end_pos        NUMBER;
  n_start_pos      NUMBER := 1;
  n_delimiter_pos  NUMBER;
  n_seek_pos       NUMBER := 1;
  n_tbl_index      PLS_INTEGER := 0;
  n_return_counter NUMBER := 0;
  v_return_string  VARCHAR2(32767);
  TYPE tbl_type IS TABLE OF VARCHAR2(4000) INDEX BY PLS_INTEGER;
  tbl tbl_type;
  e_no_delimiters EXCEPTION;
  v_string VARCHAR2(32767) := i_string || i_delimiter;
BEGIN
  BEGIN
    LOOP
      ----------------------------------------
      -- Search for the delimiter in the
      -- string
      ----------------------------------------
      n_delimiter_pos := instr(v_string, i_delimiter, n_seek_pos);
      --
      IF n_delimiter_pos = length(v_string) AND n_tbl_index = 0 THEN
        ------------------------------------------
        -- The delimiter you are looking for is
        -- not in this string.
        ------------------------------------------
        RAISE e_no_delimiters;
      END IF;
      --
      EXIT WHEN n_delimiter_pos = 0;
      n_start_pos := n_seek_pos;
      n_end_pos   := n_delimiter_pos - n_seek_pos;
      n_seek_pos  := n_delimiter_pos + 1;
      --
      n_tbl_index := n_tbl_index + 1;
      -----------------------------------------------
      -- Store the segments of the string in a tbl
      -----------------------------------------------
      tbl(n_tbl_index) := substr(i_string, n_start_pos, n_end_pos);
    END LOOP;
    ----------------------------------------------
    -- Prepare the results for return voyage
    ----------------------------------------------
    v_delimiter := NULL;
    FOR a IN tbl.first .. tbl.last LOOP
      IF a >= i_occurance AND n_return_counter < i_return_instances THEN
        v_return_string  := v_return_string || v_delimiter || tbl(a);
        v_delimiter      := i_delimiter;
        n_return_counter := n_return_counter + 1;
      END IF;
    END LOOP;
    --
  EXCEPTION
    WHEN e_no_delimiters THEN
      v_return_string := i_string;
  END;
  RETURN TRIM(v_return_string);
END;