Peer-review in the year 2000
I can go back as far as 1995, when I started my master's thesis, but if you want to dig like an archaeologist, you should probably go back a bit more.

In fact, in 1995, when we submitted my first work to a conference, we already did it by email (we submitted a camera-ready abstract). 

For journal papers, in those years, the peer-review system transitioned from using the ordinary mail to the electronic one, and at the time there was an intermediate solution (at least in my field): you would submit the papers by email and you would receive the reviewer's remarks by ordinary mail (if I recall correctly: you know, memory sometimes fail us dinosaurs!). The peer-review process was managed by the editors by email. Below, you can find a picture of a handwritten reviewer's form which I received, by ordinary mail, in 1998 during my PhD (sadly – at the time – a rejection!).



Manuscripts should have been typeset in a one-column double-spaced format, to allow the reviewers to write corrections between the lines. Note, in fact, point 13 from the form above which says: "I have marked the manuscript and am returning it with this review". Some journals still require this format (which I don't like) nowadays, even though no one sends back corrections in this way anymore.

A few years later (probably around 2000), the first websites for the electronic management of submissions started to appear. I recall that at the beginning these were very fragile systems, which frequently crashed near submission deadlines, when the number of connections increased dramatically.

I don't know what was like before those years, but I suspect that a mixture of phone and mail communications was common.

Notes


As an aside, it might be interesting to note that, at the time, we didn't have personal email addresses for all the group members: there was an email address for the PI and one email address for all the others (indeed, the common email address allowed the group members to know each other's private facts). It took a couple of years more to have a personal email address for each person at my university.
Before the advent of emails, if a researcher wanted to ask for a copy of a paper (while reviewing a paper or for any other reason), they would have sent a letter. Some institutes had preprinted postcards exactly for that purpose. Here is a picture of such a postcard that I found in the drawer of my desk, a relic from another era.