I've been 24x7 on call since the '90s. The only time I haven't been on call in that whole range is when I'm on a plane or I'm somewhere that has no service. I've replaced a bank of data servers from a campground, I fixed crippling production problems from an airplane.

Unfortunately the answer to your question relies mainly on your personal ability to fall asleep. And your ability to stay asleep.

From the time where you're actually under you need about 45 minutes to do any good sleep cycle wise. If you don't make it to REM it does you no good mentally. Making it an hour and a half is my personal minimum optimal. It should always be somewhere around a multiple of 45 minutes, obviously it gets front padded by how long it takes you to fall asleep.

I'll take an hour and a half over nothing. I can sleep in full daylight, I can sleep during thunderstorms tornadoes incredible amounts of sound around no problem.

Depending on the status of whatever I dealt with at night, I'll either take the rest of my time in the morning or at night. Usually if I was up till 4:00 fixing something I need to be up at 9:00 when people start coming in to make sure it's okay. I'll either take a few hours at lunch or cut off early for the day. I don't mind missing a few hours of sleep here and there but if it drags on for more than a couple of days I eventually have to pay the piper.

I do have another really weird sleep thing though. If it's late and I'm driving and I'm starting to doze. If I pull over somewhere and sleep for 15 minutes, when the alarm goes off I no longer have the uncontrollable urge to doze off. It doesn't do anything for my mental acuity. It doesn't make me feel rested. It just clears that insatiable need to close my eyes and shut down, at least for a few more hours.