Could they make the trip using Apollo-era tech? I believe, yes.
I'd hate to be the schmuck who went because Apollo-era tech probably couldn't create the lavish lifestyle environment you see in The Martian. Thus, it would be cramped. Really cramped. Especially given the amount of food, water, and fuel that would need to go with them. But I don't see anything about the tech that intrinsically says "no, can't be done." Thus, yes.
And despite how cramped and ugly life would be like, I expect they'd be beating away people who would want to go with a stick. I'm just sayin'.
The problem isn't the travel. It's the layover.
Here's your biggest problem:
The lowest energy transfer to Mars is a Hohmann transfer orbit, which would involve a roughly 9-month travel time from Earth to Mars, about 500 days (16 mo) at Mars to wait for the transfer window to Earth, and a travel time of about 9 months to return to Earth. This would be a 34-month trip. (Source)
Note that there's a citation request for that 500 day period. But in a sense, the amount of time on the planet doesn't matter. Read on, MacDuff!
It's not the travel time that's the problem. Yes, a plethora of problems could go wrong and since one doesn't have that big lavish ship to comfortably travel in, there's a high risk of problems during that 34-month flight (because the main travel capsule or pod cannot land, IMO, and still be believable for Apollo-era tech. Nor is the landing particularly a problem because, technically, if you launched everything in stages, you could haul an entire Saturn-V rocket and temporary gantry to get you back off the surface.
Why do I say this? Because getting everything you need into space from Earth, where the entire technological and industrial resources of at least one superpower would be available to you, is just a matter of economics (which is a deal-killer, BTW) and time. It's the other direction where you don't have a tech/industry base to draw from that's the problem.
The problem is the 16-months on the planet
Just as it's possible to haul an entire Saturn-V and gantry with you to get you back into space, it's possible to haul all the food and water you need.
And here's where MacDuff comes into it. If you can haul an entire Saturn-V with you, you can haul all the fuel you need to leave when you want, eliminating the need for an energy-conserving Hohmann window. But what would be the fun of that?
But I'm having trouble swallowing that any economically-practical habitat could be hauled with the mission that could keep people alive for 16 months. (And I'd once again hate to be the schmuck who had to stay 16 months in the command capsule.... Computers weren't that good in the 60's.)
If you can haul an entire Saturn-V, couldn't you just haul all the concrete and metal you need? (Of course, if you can haul an entire Saturn-V with you, maybe it doesn't matter if the original command capsule is abandoned... good for that crew member!)
And that's why I'm having trouble with the landing part of this scenario. We've loaded so much mass onto the outbound mission that it's become unreasonable to believe the mission could possibly economically succeed. Citizens were complaining about the cost of the U.S. space program before Apollo even came along. By Apollo 12 the average citizen was done being interested. By Apollo 13 even Congress was wondering about the cost.
Can you imagine the cost of schlepping an entire Saturn-V into space? And then an entire habitat? And everything else you'd need?
Worldbuilding is about simplification
Having said all that, ignore it. Too many worldbuilders, in the quest to be "as realistic as possible," forget not only economics, but forget that worldbuilding is about simplification. Godlike accuracy doesn't make for a better world. It impresses some people, but most find that level of detail boring.
Ignore everything I just said and go build your alternate timeline. It's believable enough.