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When I was taking my one (weak) Russian course years ago, I ran across an MP3 with sample sounds. The pair that got me very worried was Борис vs Борись. At the time, I could hear only the vaguest difference between those sounds, much less be able to reliably speak them. Recently I was wondering how common it is in Russian that two words differ only by a soft sign. Is that rare, not uncommon, or happens all the time? Is there any list of words like this somewhere? I was thinking of developing a list and saying one of the pair to a native speaker and asking them which one I said, and having them say a random choice and me telling them what I heard.

I remember reading about how close the French words for "over" and "under" sound (both sound like "desous"), and that French orators tend to avoid using them because it is hard to make sure everyone hears the right word over a PA system, etc. Does the same thing happen in Russian, or is the difference really obvious to the native Russian ear? Or does Russian encode words differing only in a soft sign with wildly different meaning so the correct word is heard easily in context? (Apparently, Борись is archaic.)

These days when I meet a native Russian speaker, I often ask them to pronounce Борис vs Борись, so I can again marvel at how subtle the difference is to this native English speaker's ear.

The vowels in dessous and dessus are clearly different - like u in full vs u in dude. Unless you pronounce dude like dood there shouldn't be any confusion between 'below' and 'above' in French.

There is a clear difference to a Russian speaker between the consonants with and without the soft sign. Конь sounds different from кон and даль sounds different from дал.
Palatalized vs non-palatalized consonant is easily audible for a native speaker. There might be more combination like that, but it's the only one I can think of that comes up often.

In fact the only two words that sound very similar and often have to be repeated loudly in Russian that I can think of are Июнь and Июль, but it's because of l and n sounding similar in that particular combination, especially when pronounced quickly.

Try saying Ron, then Ronnie. And try to cut off the 'ie' from Ronnie while keeping the trailing soft sound after the 'n'. It sounds as if you were going to pronounce 'e' or 'i' after a consonant but changed your way at the very last moment and the vowel got cut off, leaving consonant sounding different than normally.

Find an audio of дельфин In Russian. Don't worry about the stress (it's on the second syllable), just listen to the l sound and compare it with l in English dolphin. You can use Google Translate, the pronunciation is alright there. It might not be clear if you're not used to the sound, but once you get it, you shouldn't be confused afterwards.

This video starts right at chorus.
https://youtu.be/FqB-m0dKO-g?t=78
At the second line (repeated throughout) there are two words with and without soft sign after т - память and лет. The singer cuts off 't' very nicely so лет has a very nice short hard т sound, (while память has a soft t).

It seems to me that there are not so many Russian words like that.

Here are a few pairs I can think of:

Мол - моль

Ролл - роль (the spelling differs a lot but the sound at the end is very similar - the only difference is the l' sound at the end of роль)

Кров - кровь

Вонь - вон

Коп - копь

Зор - зорь

Мороз - морозь

Of course, there may be more examples. If I remember some more, I promise to edit my answer adding the pair(s).

As for understanding which word is being said, I don't think it's too hard. I can't remember a time when I or someone I know had such problems.