Q:

How could letters overlap in manual typesetting?

Generally speaking, ascenders and descenders should not touch each other in the text body.
This is self-explanatory as it could lead to text that is difficult to read.
However, Erik Spiekermann points out that there are use cases that profit from partly overlapping letters, e. g. to render headings more forceful (Spiekermann 1986: 43).
He also provides an example (Spiekermann 1986: 42, arrows by me), which says:

There’s a rule according to which descenders and ascenders must never
  touch. There’s an exception to this rule which states that they may
  touch if it looks better.

In this example, the letter g touches the letters ü and R in the following lines.

In digital typography, letters are not necessarily connected to the size of their “metal” block (which still exists virtually) anymore, e. g. in the font Amsterdamer Garamont, the lowercase h exceeds its block on the top while the lowercase p exceeds its block on the left and bottom side (Forssman and de Jong 2014: 86). The authors write (transl. by me):

In manual typesetting, this would not be possible; the overlapping
  parts of the letter would collide with the letters on the lines above
  and below, and break.

The question:
Before the invention of digital type, in manual typesetting with lead body type, how were overlapping letters produced, and how were the problems described above solved?

Works referenced:
Forssman, Friedrich and Ralf de Jong. Detailtypografie. Mainz  2014 (2002).
Spiekermann, Erik. Ursache & Wirkung: ein typografischer Roman. Erlangen 1986.

A:

Not all typesetting before the digital era was done using moveable metal type. In 1986, phototypesetting was already available, and had been since the 1950s.
Phototypesetting involved projecting letter forms onto strips of photographic film, for reproduction in offset lithography. The lines of text could then be cut and pasted onto the copy, and gaps between joining letters could be inked manually with a pen, then a negative could be imaged from the copy, then a printing plate made from that. 
Example of how page layout and typesetting was done, before digital, but after the introduction of phototypesetting

And before you ask, yes I am old enough, and indeed it was how I learnt to do layout - just as digital methods began to be adopted widely within the industry in the latter half of the eighties and early nineties, and certainly by that time moveable metal type was viewed as a rather old fashioned technology.

A:

Even when talking about metal type, letters were not all square. Take a look at this "f" for example, and you can see a bit that's protruding, and would make it overlap with the next character.

A:

I agree with @BillyKerr -- 1986 was at a minimum movable type and Photo Mechanical Transfers (PMT). In other words, absolutely hot type and not cold type. Also realize it was the early 80s when computers were starting to be in use. In the later 80s they were already becoming a staple of many workflows as the software started catching up. Photoshop, Illustrator, Freehand, Pagemaker, and QuarkXpress all existed by 1987. So it's also not impossible that the type was set on a computer in 1986.
But... if it had to be done with metal type..
A Double-Hit or Multi-Pass run would work
Remove the Gs.. print it.. put the Gs back, remove everything else.. hit it again.
Or just run every-other line once, then run the other lines as a second hit.
Printing presses are not restricted to doing everything in a single pass. In fact it is still very common today to see a multi-pass press run. Often varnishes or gloss is applied as a second pass, but you could just as easily run a second pass of ink. Some smaller offset presses only have 1 or 2 inkwells so they have to run a multi-pass system to print anything more than 1 or 2 colors.

It's impossible to tell in your attached image because it's a digital image, but when looking at a printed price on paper, in real life, not digitally.... with a linen tester/loupe... you can often see that the ink may be a touch more dense in the overlap areas... which would indicate a multi-pass of the ink.

