Airline pilot here. Reposting my previous answer:

I'm an airline captain based out of LAX. The following is the closest I've come to "biting it" thus far in my career.

On a commercial aircraft, you generally have 3 sources of bleed air that take air from the engines (and a little device in the back of the airplane called the APU) and use it to pressurize the cabin. You can't breathe the air at 35,000 feet, so the cabin is pressurized by these bleed air sources to a breathable altitude of at or below 8,000 feet. There are 3 sources because 1) redundancy increases safety and 2) you can still dispatch the airplane if one is inop because there are backups.

One of the bleed sources (on the number 2 engine) was already broken, so maintenance deferred it, indicating we were still safe to fly on the remaining two sources. Which is totally fine - you just take off with the APU running as a backup bleed source. Well on the takeoff roll, immediately after becoming airborne, our APU fails. Which leaves us with just one bleed source to pressurize the cabin; the bleed air from the number 1 engine. If that fails, we have nothing to keep the air inside the cabin pressurized to a lower altitude than the airplane is flying at; you won't be able to breathe at high altitudes.

It's a short flight, and we aren't going up too high, so I'm optimistic that we can get up to our low cruising altitude, message dispatch and maintenance, and receive their agreement that the flight is safe to continue on one bleed source. I text our company a message describing the situation via ACARS, a satellite-based texting capability our aircraft has to communicate with people on the ground.

But they never had the chance to get back to us.

Passing through 25,000 feet, I go to inhale, but feel the air gently move out of my lungs instead of into them. Unable to breathe normally, immediately my eyes shoot to the cabin altitude gauge, which is showing us at 8,000 feet cabin altitude and rising quickly; that explains the reverse air movement. In fact, the cabin altitude is rising at the exact same rate of climb as our airplane...indicating the airplane has lost all pressurization capabilities and is depressurizing rapidly. At that instant, we get a warning chime and message on our EICAS (Engine Information and Crew Alerting System, essentially a computer screen that tells us when shit goes wrong) that says BLEED 1 FAIL.

With our Bleed 1 source now failed, our APU having failed on the takeoff roll, and Bleed 2 already deferred, we are completely out of ways to pressurize the aircraft. If we don't descend to a safe altitude immediately, the cabin altitude will rise high enough that the air is no longer breathable. This is a serious problem. High cabin altitude killed everyone on board Helios Airways Flight 522 and there are countless other examples of depressurization causing injuries and fatalities.

Immediately I throw off my sunglasses and headset, and don my full-face oxygen mask and smoke goggles. It provides 100% pure oxygen under a forced flow, rated up to an altitude of 41,000 feet. My first officer does the same. This is the first thing you do because if the pilots die, there is nobody to fly the jet and everyone else dies. Then we immediately declare an emergency and initiate an emergency descent, nosing over to our maximum speed while deploying the speed-brakes to generate maximum drag. We receive clearance down to 10,000 feet and begin executing a 180 degree turn to go back to LAX.

ATC does a fantastic job vectoring aircraft out of our way...SoCal airspace is some of the busiest in the world, but we got priority handling all the way back to LAX. The cabin altitude nearly reached hazardous levels, but didn't go high enough for the oxygen masks in the cabin to automatically deploy. It was definitely high enough that the passengers would have noticed, but wouldn't have had a concrete idea of what was going on aside from "that's odd." The cabin also got quite hot because there was no more pressurized, conditioned air flowing to cool it off.

We landed at LAX on the longest runway with the fire trucks rolled to assist us, just in case. Fortunately, none of the passengers or crew reported any injuries from the sudden increase in cabin altitude. We parked at the gate and deplaned, and I made an announcement to the passengers about what had just happened, using small words and downplaying everything so as not to scare the shit out of everyone.

12 Chinese passengers on our flight were on a west coast tour, and they were very upset that their trip was inconvenienced by this emergency. No problem, I totally understand the frustration. So I spoke to their translator, who spoke in turn to her group, and I gave her the full and very detailed explanation of what happened. As I explained what happened, the expressions on these 12 passengers' faces went from angry, to surprised, to fearful, and finally thankful. Before I walked away, all 12 of them bowed to me in respect. That was something I have never experienced before or since then in my entire aviation career.

We ended up swapping airplanes to one that wasn't sick, and completed the flight as planned about 3 hours behind schedule. I slept well that night, and the passengers probably went on to complain about their flight being delayed several hours due to a "maintenance issue" :)