The Big Apple Hostel was at 119 West 45th Street, the the heart of Times Square in Manhattan. It opened in 1994 and closed in 2014, when the building was sold for $12 million.
   
The location is now the Merrion Row Hotel and Public House.
 
         
Big Apple Hostel (archived, January 28, 2014)
Big Apple Hostel - 119 West 45th Street - New York, NY 10036 - Tel. 212 302 2603 Fax 212 302-2605 -
       
Merrion Row Hotel and Public House
119 W 45th St, New York, NY 10036
212 461 4920
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Our Beaux-Arts building, built in 1920, was one of Times Square’s first hotels. Frequented for decades by local actors and artists, as well as families in search of a truly New York experience, it has been reborn. A deep nod to the building’s rich history, Merrion Row remains a paragon of contemporary luxury.
   
Google Groups: rec.travel.usa-canada
HOSTELS: NYC. $12 - $22
Jim722
Sep 29, 1994, 11:17:02 PM
NYC has a reputation for being expensive but it doen’t have to be the case. NYC has a large number of very good hostels which can suit almost any taste and any budget. Also, it is very easy to get from the airports. For instance, from JFK, there is a free bus to the subway system, then the subway costs only $1.25 to get into Manhattan. The following is a list of hostels taken from The Hostel Handbook for The USA and Canada:
BIG APPLE HOSTEL, 119 West 45th Street, NYC 10036 (212)302-2603 $17
 
Google Groups: rec.travel.usa-canada
HOSTEL NEWS: DECEMBER 94
Sugar722
Dec 6, 1994, 6:35:14 PM
(...)
BIG APPLE HOSTEL, 119 West 45th Street
New York, NY 10036 (212) 302-2603 AAIH $17
               
Google Groups: rec.travel.usa-canada
Cheap guide to NYC - Here It Is.
.(JavaScript must be enabled to view this email address)
Jul 16, 1995, 3:00:00 AM
(...)
Big Apple Hostel 302-2603 $17+Tax MT
 
Google Groups: bit.listserv.travel-l
NYC budget accommodations
Dennis Taylor
May 14, 1996, 3:00:00 AM
(...)
Big Apple Hostel $20
119 W. 45th Str. (between 6th & 7th Ave.)
302-2603
 
Newspapers.com
28 March 1997, News-Press (Fort Myers, FL), “Doors shut at 47th St. Photo” (Associated Press), pg. 8A, col. 3:
“I used to send people there all the time. It was very reputable,” said Tony Santiago, an employee of the Big Apple Hostel, which lodges young international travelers and is right next to the 45th Street store.
 
Twitter
Andy
@torak
Replying to @hargitaybensons
@gabylevin Big Apple Hostel! http://www.bigapplehostel.com/ amazing rates, fantastic location, actual accom is good enough. Perfect
5:41 AM · Feb 8, 2010·Twitter Web Client
 
Twitter
Web Travel
@web_travel
Big Apple Hostel New York City: The Big Apple Hostel is located in the heart of Manhattan.  Literally, it is only ... http://bit.ly/c7nCGr
8:43 PM · Aug 1, 2010·twitterfeed
     
Commercial Observer
Big Apple Hostel Sells for $12M
BY LAUREN ELKIES SCHRAM
MARCH 31, 2014 1:05 PM
Big Apple Hostel, at 119 West 45th Street, has sold for $12 million and is slated to become a tavern and hotel or hostel, according to city records and the buyers’ attorney.
 
Yelp
Big Apple Hostel
Yelpers report this location has closed.
119 W 45th St
New York, NY 10036
(...)
Sam J.
Plano, TX
5/7/2015
What happened?! I stayed here once or twice years ago… It was a great little hostel RIGHT next to Times Square. Definitely middle-of-the-city. That’s too bad it closed down… I always had it as a backup in my mind if needed. :’( Come to think of it, I remember walking by their location last time I was in NYC and didn’t see the sign… Guess I just didn’t want to accept the truth.
 
Edit: Holy…. I just saw why they closed: “Big Apple Hostel Sells for $12M” on 3/31/14. I’d sell it, too!