One possibility is that the meat was cooked in a crock pot.
This would make sense on a few levels.
First, since it is a hospital and presumably they have a large amount of people to feed, cooking multiple steaks in a large crock pot at once would save time and still produce quality food.
Second, cooking the meat slowly in a crock pot (or in liquid) over the period of 10 hours will produce incredibly tender meat, especially if the steak was already of good quality to begin with.
Similar process is used to produce various roasts and also pulled pork, for example, which will literally fall off the bone once done cooking.
cooking.stackexchange.com/users/18418/Travis J
First of all: this is the first time I've seen anyone ask to recreate hospital food...
I'm happy your dad didn't have the same experience most people do with hospital food.
One technique that can give meat that tender is velveting the meat.
This is a Chinese cooking technique where the meat is marinated in egg white, wine and corn starch before cooking.
This will give a velvety smooth texture to the meat, and it will basically melt in your mouth.
cooking.stackexchange.com/users/31313/Johanna
This is my recipe for a sublimely tender roast: Grab a large chuck roast or brisket.
Start in the evening by smoking or roasting the meat at 225 F until an internal temp of ~150-160 F is reached.
Place the meat inside of a covered roaster on a metal rack (important)
then place in the oven at a rendering temp (say 170 F)
overnight and all day until you get home from work.
Your oven needs to be calibrated, because holding meat at too low a temp is a safety concern (see the bottom of answer for more).
Chill over night before slicing and portioning for the most cohesive, moist and unimaginably tender product.
You could rest it on the countertop and slice the night of.
Just know that the resting time on a large roast is considerable.
You can adjust the time and temperature variables to simplify the recipe further: roast in the oven at 350 F till ~150-160 F internal temp is achieved.
Drop oven down to 170 F in a roaster till tender.
The long answer: (Nerd Alert)
Achieving the perfectly tender roast is a complicated subject because the temperatures at which different meat tissues break down are complicated.
Above 140 degrees F
you are out of the food safety danger zone.
This means you can take your sweet time as long as you are above this mark
(this is a broad generalization, see the bottom of answer).
At around 160 degrees, collagen and fats begin to break down or 'render', causing meat to become tender and juicy.
However, there is a point where the proteins will begin to tense up and toughen (Don't know this off the top of my head, but the hotter you go the tighter they get).
There is a special temperature zone where you can achieve full rendering (maximum tenderness) without encountering dryness.
This is achieved by getting the meat out of the danger-zone then holding it within the rendering zone for a very long time, without leaping into the tightening zone.
The reason that a most pot roasts are fall apart tender but dry is because the typical stock pot will eventually simmer on low.
Water boils at 212 F at sea level, at which point most meats are fully rendered but dry.
Safety
Adendum:
Food safety is a matter of temperature and time.
The FDA publishes charts that tell you how long a piece of meat has to be held at a given internal temperature to be safe to eat.
To say that beef is safe as long as it is over 140 F internal is not strictly correct, as time is also a factor.
cooking.stackexchange.com/users/33792/Derpy
What you describe sounds a lot like swiss steak, in which you use cube steak (an inexpensive cut that's been mechanically tenderized).
You sear the cube steak, then slow cook it in a flavorful sauce until it's extremely tender.
cooking.stackexchange.com/users/67/Joe
Maybe the meat was made with transglutaminase, a clotting agent that has the ability to bind protein-containing foods together.
As I understand it, the results are very tender, often being served as filet.
This from http://www.mirror.co.uk/news/weird-news/you-eating-fake-steak-customers-5856558
I have been served some very questionable filet, soft, mushy, and flavorless.
Though the guys at https://www.youtube.com/watch?v=ZhgOEsAd1xY say the glued together meat scraps cook up and taste like filet.
Watch the taste test.
cooking.stackexchange.com/users/45403/Wyncia
Being a hospital, where food safety is even more important than otherwise, I'd imagine they step away from the more elaborate methods of tenderizing meat.
The first thing I thought of was the industrial version of slow cooking.
The may have used something like this Alto Shaam cook and hold oven : Manufacturer touts:
Halo Heat tenderizes even the toughest cuts of meat by activating their natural enzymes
Many restaurants use the above to slow cook products over night at low temp.
It yields better results than most home methods.
I've seen some pretty great product come out of those.
If the hospital used this, and I suspect they did, you'll have to try something like @Chee's Burgers answer.
(if you do, check the oven temperature with an oven thermometer for safety reasons)
Mechanical tenderizing is a possibility.
You can get home use tenderizers like these (I like the one on the left):
And... there is always the possibility the hospital purchased meat that had gone through some special processing.
I wish you had a picture of the cut :)
cooking.stackexchange.com/users/21367/Paulb
This sounds similar to something my mother used to cook, she would get oyster blade and put onion and gravy powder on top before wrapping it in tin foil and putting it in the oven.
The result was a tender piece of meat with a thick brown gravy on top, it tended to be more on the salty side due to the use of pure gravy powder, was the meat salty?
cooking.stackexchange.com/users/80400/chris williams
My dad was recently in the hospital for intestinal surgery.  While he was recovering they served him some beef that he was very surprised to find incredibly tender.  He let me have a bite.  It was so tender you hardly needed to chew at all.   It was served as some kind of boneless fillet with a dark brown gravy on top.  I've cooked a lot of beef, but never was it this soft.  Can anyone help me find this recipe or teach me a technique with the same effect?
Extremely tender beef at hospital
cooking.stackexchange.com
<meat><beef><tenderizing>
cooking.stackexchange.com/questions/55351
cooking.stackexchange.com/users/33947/Sam Washburn
The meat might have been cooked for a long time in a crock pot or made with transglutaminase, a clotting agent that has the ability to bind protein-containing foods together. A similar result can be gained by putting onion and gravy powder on top of oyster blade before wrapping it in tin foil and putting it in the oven.    Three methods of producing very tender meat are cooking in a crockpot, cooking in foil in the oven and using transglutaminase.
Meat cooked in a crockpot is usually very tender. Transglutaminase produces very tender meat. Cooking meat in the oven wrapped in foil produces tender meat.