So George; I bet you think that only 1980 Chevys came with a 5.7 liter 350 engine, or else that all 1980 GM 350 5.7s are the same; and that's why you didn't bother to tell us what brand of car you have.
Unfortunately; GM also used a 5.7 liter 350 engine in the 1980 Buick Electra and Riviera; in the 1980 Pontiac Bonneville and Catalina; in the 1980 Oldsmobile 98, Custom Cruiser, Cutlass Delta 88, and Toronado; and in the 1980 Cadillac Seville. And there was also a 5.7 liter 350 diesel engine used in most of those brands. Some of these cars use different parts in their engines or electrical systems; even though they're all GM 350 5.7 liters. Is this a good reason for us to want you to tell us what brand of car you have???
If your 1980 350 5.7 gas or diesel car or truck has an automatic transmission; there is a part called the neutral safety switch, which cuts off power to the starter unless the shifter is either in Neutral or Park. There are 2 independent sets of contacts in that switch; and the Park contacts wear out sooner than the neutral contacts, because they are used less often. So if this has happened to your vehicle; it might start if the shifter was moved to the Neutral position and started there. If that makes it start; the neutral safety switch should be replaced. If your 1980 350 5.7 gas or diesel car or truck has a manual transmission, there is no neutral safety switch.
The other part that can cause this problem is the ignition starter switch. The start contacts can fail; even though the rest of the switch works.
Some tests of the starter solenoid just check whether it pulls in when energized; but do not check wther the contacts conduct power through to the starter. If your solenoid contacts have burned out; it may not be sending power to the starter; even though the solenoid pulls in.
And a starter motor which has the brushes worn down close to the limit may run in a bench test; but will not run when it has to turn the engine.
A battery that is grounded to the engine; but not grounded to the body with a second ground wire, will be prone to not energizing the starter solenoid at times. Similarly; if the original ground wire between the engine and the firewall has been removed; that also can cause this type of problem. There MUST be an auxiliary ground wire from the battery negative terminal to a bolt in the inner fender; and another such wire from a bolt in the engine to a bolt in the firewall.
The main battery ground cable MUST be connected to a bolt that threads directly into the engine block. It cannot be connected to a bracket; nor to a non-metallic part, nor to any part except the engine. If the existing battery ground cable is too short to reach the engine block; it should be replaced with one that reaches.
And both battery cables and battery posts have to be scraped shiny clean on their contact surfaces, and tightened to the point where they cannot move from hand pressure.