Actually, we had these arcane things called "newspapers" that were filled with hundreds and hundreds of ads for apartment/house rentals, sales and even shares.
You figured out what you could afford and went through these arcane things and marked the ones that interested you. Then you gasp made a phone call to find out:
1. If the place is still available;
2. When can you come to look at it;
3. If you like it, snap it up (by writing a check[0]) before someone else does.
As for brokers, the ads generally noted that it was 'no fee' if the rental was direct with the landlord. Apartment shares were generally 'no fee' as well, since you were just moving in with roommates you didn't know.
I had some really weird experiences looking for both shares and apartments back in the 1980s and 90s. But it wasn't all that difficult or that much more time consuming than using the real estate websites these days. They're essentially the same thing, except with more photos.