I found Times Square House on Hostelbookers.com, and after reading glowing reviews on both Hostelbookers and Tripadvisor (though one of the tripadvisor reviews is a misplaced review for another hotel and several are in a language I can't read) I was excited to try it out. I wanted to contact the hostel to inquire about their "no luggage storage" policy (does that mean my luggage won't be secure in my room when I'm not in?) and their cash-only policy (what would the total cost be, with taxes?), but couldn't find contact information anywhere on the web... In fact, I couldn't find any record of this hostel elsewhere on the web because, it is, in fact, not a hostel, but a (very nice!) man renting bedrooms out in his Hell's Kitchen apartment.
What this means for you, staying at Times Square House: 1) there's no "staff," no employees to speak to in advance about your concerns, or during your stay. The apartment owner is available via e-mail AFTER you book, and in-person during your stay, but keeps regular business hours (meaning, he goes to sleep in his bedroom across the living room from you or downstairs from you at night, and isn't up till morning, so there's no assistance after hours.
2) There are four bedrooms with up to two people per, and two bathrooms total. So at 11 AM checkout, you will have to battle the other occupants for a bathroom like you're back in your college dorm. My traveling companion and I managed to check-out in time, so no idea if there are consequences if you don't because some other occupant wants to give herself a makeover in one of the only two bathrooms present.
You also have to deal with the other occupants' hygiene issues, like clumps of hair in the drain, toothpaste in the sink, wet towels on the floor, etc. Unlike a real hostel, where you'd have many restrooms to choose from, and cleaning staff making the rounds, you're stuck with whatever condition the previous person left the bathroom in at Times Square House.
3) Speaking of the bathrooms, both at Times Square House feature signs that read "please don't use too much toilet paper, toilet clogs easily!" My traveling companion experienced this first-hand, and, with no plunger present, well, you can imagine. Again, no hotel or hostel would allow plumbing in this condition.
4) You can get a lovely, private room at Times Square House. While the room was not the one shown on the hostelbookers site with the exposed brick, it was very nice. Hardwood floors, an AC unit, cool wall art and a comfy bed. No TV, but that was ok. The bed only featured a bottom sheet and comforter (no top-sheet, and I'm guessing the comforter doesn't get washed after each stay, so lots of potentially-commando travelers+direct contact with comforter=EW!). My companion and I paid $120/night to stay at Times Square House, which is reasonable for a private room, and tough to find within Manhattan proper.
5) This "hostel" is in Hell's Kitchen, NOT Times Square. I thought such a pricey hostel, named Times Square House, whose listings feature lots of pics of Times Square, would actually be in Times Square. I was willing to pay double what other hostels were charging per night to be right in Times Square. My companion and I walked from Times Square station to Times Square House, and it was a sweaty, lengthy hike. But more importantly, the neighborhood, Hell's Kitchen, is sketchy. And dirty. Like a lot of NY. But there are a lot of convenience stores/little produce markets. And Times Square House is right next to the Hell's Kitchen Flea Market, which was super cool! Lots of neat finds, for pretty good prices.
6) Since Times Square House is literally someone's house, and he lives there, it's quite awkward at times. You're bathing in someone's shower with them, a complete stranger, right next door. The owner is generous and offers access to the items in his kitchen, but using a stranger's dishes, drinking his milk, brewing coffee in his coffeepot just felt too weird to me, with him there. I was concerned about inadvertently breaking a rule (no shoes in the house, no cooking in the kitchen, shoes go on the rack when you're home, no alcohol in the rooms), using someone's "special" mug or the last of his girlfriend's soymilk. Maybe I'm too neurotic for this kind of setup. Love the anonymity of a real hostel or hotel! Again, the owner is very nice, and did his best to make Times Square House comfortable.
7) Bring your own soap/toothpaste/washcloths etc. A towel was provided, but there weren't any in the bathrooms themselves, nor soap in the shower, nor tissues in the bathroom or bedrooms, and while the toilet paper did get refilled when it ran low, I didn't see any extra rolls available in the bathroom.
Overall, Times Square House is pretty bare-bones, and not quite what I was expecting. But it did the trick as a place to sleep after adventurous days in NY. If that's all you need, and don't mind the close-quarters, go for it!