My Dad played high school and college football in the 1930s. He and the other starters played both offense and defense. What prevented a team from putting out 11 fresh players at once when the ball changed sides?
Wikipedia covers this reasonably well.
In the first era (what you're talking about, basically) of the one platoon system, there weren't really rules about substitutions, but there was a major limitation: the size of the team. You didn't have 80 player teams back then. In the 1940s, a few teams (notably, Army) started the two platoon system (what we have now); but even before that, some used substitutions (for example, Knute Rockne). So during your dad's day, it was a combination of financial reasons, athletic talent, and simply the fact that it wasn't done. (Remember, this was still somewhat of a "gentleman's game" at the time, primarily a college game, and so things like "honor" were very relevant to the coaches and players.)
Then in 1953 this was outlawed, allowing only one sub per play, partly in order to reduce the cost of fielding a team. This was changed in 1964 to return to allowing two-platoon football, which is what we basically know today.
According to this page of NCAA rule changes, the following rules changes occurred over the years. I cannot substantiate all of the individual details, but the ones I can (1941, 1953, 1965) are consistent with what I've mentioned above.
1876--Fifteen players to a team and few if any substitutions.
1882--Replacements for disqualified or injured players.
1897--Substitutes may enter game at any time at discretion of captains.
1922--Players withdrawn during the first half may not return until the second half. Players withdrawn during the second half may not return to the game.
1941--Players may substitute at any time but may not be withdrawn or the outgoing player returned until at least one play has intervened. "Platoon" football made possible.
1948--Unlimited substitution on change of team posession.
1953--Two-platoon abolished and players allowed to enter game only once in each quarter
1954-64--Changes each year toward more liberalized substitution and platoon football.
1965--Platoon football returns. Unlimited substitutions between periods and after a score.
1974--Substitutes must be in for one play and replaced players out for one play.
1993--Players who are bleeding or whose uniforms are saturated with blood must come out of the game until their return has been approved by medical personnel.