Space faring would be possible, reaching an other star system would be a far stretch and require centuries of industry development at 1980's tech level.
A lot of technology's we think of as new were actually pioneered in the 60's and 70's. For example the molten salt thorium reactors were build and tested first in the 70's before the project got closed down because of budget constraints and regulations.
Some high performance materials existed like silicon carbide and hafnium carbides and oxides. But now we can make them with an higher purity and better.
Lasers, radar, magnetrons and basic computers all existed.
A lot of NASA papers regarding hyper-sonic space-planes and ionic propulsion drives are published with a date from the 60's. So google around for those.
The civilization needs to be incentivized to go all in on space technology and to put in the capital cost of building a permanent space industry. The politics in our current timeline were not conductive for that, and the green movement effectively killed all development on nuclear fusion.
On the propulsion site we have LRP1 and Hydrogen oxide rockets to get mass into space and certain type of ion or plasma drives are possible in combination with solar power or nuclear power to slowly accelerate spaceships to high speeds for interplanetary travel.
Look into old designs for the space-shuttle to see what engineers envisioned and though of as possible before the over engineered monster what we know now as the Space Shuttle got build. There are some awesome designs to get onto space quick.
There are some single-stage-to-orbit concepts which probably would have never worked, so if you go for realism steer away from those. But there were some projects that managed to land booster rockets SpaceX style, so that was clearly possible. And reusing rockets would be a prerequisite to bring enough 'stuff' up to start a Space faring civilization.
Further await from the Sun than about Jupiter would require a nuclear reactor for power generation. We never build them because of international treaties not because we didn't have the technology. Electric propulsion requires tons of energy, so a nuclear power source is the obvious way.
There were a lot of concepts that used tons of electricity to accelerate ions or heat up plasma's using magnetrons for electric propulsion. Few got build and tested. The political will was not there, but the technology was definitely possible. These can slowly accelerate a spaceship to very high speeds, but it would still take at least a century to reach the nearest star.
To colonize an other star would require a spaceship the size of small city. Which is possible. A steel O'Neil cylinder can be made several miles wide with technology of the time. But it would require a massive industrial complex in orbit, complete with moon bases and asteroid mining to supply enough raw material. It is not thinkable to launch that amount of materials from earth.
Some more points.
- Robots that execute preprogrammed steps are possible.
- Remote control is possible, consider light-speed related lag.
- Anything with AI requires Gigahertz computing and is not possible.
- Maybe early forms of plastic 3d printing?
- Genetic research is not possible, requires advanced computers.
- Every design is made with pencil and rulers. CAD design was to primitive to make a real difference in design style.