Select text between two patterns with awk or sed

A GNU* awk variant, just make PAT2 the record separator RS, PAT1 the field separator FS and print the last field NF, making sure that the output is not the result of a repetition of RS

awk 'BEGIN{RS="PAT2"; FS="PAT1"}NF>1{print $NF}' file1
 This is the
text that I want
to get 

 This is another text that I want
to get DONE

*Noted @EdMorton

This, with GNU sed, as ugly as it is, I think does the trick:

sed -e 's/PAT1/\nPAT1\n/' -e 's/PAT2/\nPAT2\n/' file | sed -n '/PAT1/,/PAT2/{//!p}'

Take PAT1 and PAT2 and and add newlines at beginning and end:

sed -e 's/PAT1/\nPAT1\n/' -e 's/PAT2/\nPAT2\n/'

Blalala
'Omfoem From 
balanf 
PAT1
 This is the
text that I want
to get 
PAT2
: apples

Print the text between PAT1 and PAT2:

sed -n '/PAT1/,/PAT2/{//!p}'

 This is the
text that I want
to get 

With any awk in any shell on every UNIX box:

$ awk 'sub(/.*PAT1 */,""){f=1} f{if ( sub(/ *PAT2.*/,"") ) f=0; print}' file
This is the
text that I want
to get

The above works for the sample input you provided, if you have other input in a different format that this doesn't work for (e.g. nested start/end strings or a start string after an end string on the same line) then please edit your question to show that.

With GNU grep(1)

grep -zoP "(?s)(?<=PAT1 )(.*)(?= PAT2)" file

Tests

$ cat file
Blalala
'Omfoem From
balanf PAT1 This is the
text that I want
to get PAT2: apples
Whatever: oranges

$ grep -zoP "(?s)(?<=PAT1 )(.*)(?= PAT2)" file
This is the
text that I want
to get

From grep(1) man page

-z, --null-data
  Treat the input as a set of lines, each terminated by  a  zero  byte  (the  ASCII NUL  
  character) instead  of  a  newline.  Like the -Z or --null option, this option can be 
  used with commands like sort -z to process arbitrary file names.

-o, --only-matching
   Print  only  the  matched  (non-empty) parts of a matching line, with each such part 
   on a separate output line.

-P, --perl-regexp
   Interpret PATTERN as a Perl regular expression (PCRE, see below).  This is highly 
   experimental and grep -P may warn of unimplemented features.

Regex explanation:

(?s) activate PCRE_DOTALL, which means that . finds any character or newline.

Uses a Positive lookbehind assertion (?<=PAT1 ) and Positive lookahead assertion (?= PAT2) which causes to only print the capture group (.*).

Caveats of this solution:

As stated by @bushman this only works if there only exists in the file a single occurrence of the two patterns.