The topic pretty much says it all. I was watching a game yesterday, and at some point a punt went out of bounds, probably 50 feet in the air. The refs fairly quickly spotted the ball, however. How do they know exactly where the ball left the field? It seems pretty difficult to just eyeball it, but they don’t seem to be using any tools. A margin of error of 5 yards seems pretty normal if they just guess, but that kind of difference can have a big effect on a game, so I can’t believe that they estimate… Somebody fill me in here.
They estimate. If I were the one doing it, I’d consider where the punt originated and where it lands, and extrapolate based on that line. If the ball is closer to the ground, an official standing on the sideline can watch for when the ball crosses out of bounds, and estimate a line straight down from there.
20 (or more) years of training and good guesses.
The deep official can signal (by raising his hand) that he’s guessing and needs help from the referee, in the middle. If the referee has a good idea, he’ll raise his hand, too. When he thinks the deep official has reached the right spot, he’ll make a chopping motion.
I was instructed that best way to guess is:
If the home team punted it out, go deep, and keep moving up the sideline until the crowd starts to boo. If the visiting team punted it out, go deep, and move up the sideline until the crowd starts to cheer. 
This is what I’ve always heard, as well.
The referee can stand approximately where the punter kicked it from, and look down the “flight path” of the ball. The deep official walks up the sideline, watching the referee. When the deep guy crosses the path, the ref will signal him to stop.
It’s kinda crude, but it’s a lot better than just a guess.
By “deep official”, do you mean the line judge? In rugby, (s)he’s called a touch judge, and they run along the sideline until the spot where the ball went out is right above them. Still a guess, but I like the way you were instructed to guess!
Spotting the ball on any play is really a judgement call. It’s not where the runner goes down, it’s where the ball is when the runner goes down, and that’s not always easy to see (how many times do you see a ball carrier trying stretch out for a couple of extra yards after the tackle. It’s not an exact science but for the most part, NFL officials are remarkably accurate at being able to eyeball the spots.
Actually, the system is fairly sophisticated. It’s been described already.
The referee stands in the backfield along the line of flight of the punt. The line judge runs from deep up the sideline, with his arm over his head to be seen by the referee. When he intersects the line of flight of the ball, the referee tells him to stop.
How much more high tech do you need to be? 
Technically
Field Judge or Side Judge in 7 man (7 official) mechanics (which the NFL and major colleges use) These are the guys that start 20-25 yards down field (even with the return man on punt plays) along the sideline.
Field Judge or Back Judge in 6 man mechanics (mid-level colleges, mostly). This is acutally the same positions as FJ and SJ, but there is no SJ in 6 man mech. In 7 man mech, the BJ lines up in the middle of the field, in 6 man, the on the sideline.
Back Judge (from the middle of the field) or Line Judge (from the line of scrimmage) in 5 man mechanics ( small colleges, high schools)
Line Judge or Head Linesman in 4 man mechanics (high schools in some states / levels).
For more detail than you probably need, the 7 officials are:
Referee: white hat, talks a lot, offensive backfield
Umpire: near defensive linebackers, gets in the way a lot
Head Linesman: runs ten yard chains, works on the [visitor’s] sideline, at line of scrimmage
Line Judge: opposite head linesman, line of scrimmage
Back Judge: middle of field, about 20-25 yards downfield (“deep”)
Field Judge: same side as line judge, deep
Side Judge: same side as head linesman, deep.
There is an official in the backfield of the kicking team. When he sees a punt headed for the sideline he quickly gets on the line of the kick and marks the line for himself by pointing along the line. Then another official goes to the sideline and runs down it. When the backfield official is pointing at the sideline official, voilà, that’s the spot.
Well, I see brad beat me to it. Damn.
brad?!
Oh, wait, they always say an official does his best job when no one notices him. Never mind, carry on…
Yeah, I think I have to pass the credit up a post on this one. Mine was a glorified “me too!” 