Lead-acid power is what the EV1 came out with, so something similar to that could have existed in the 1960s. Cheaper vehicles would have been closer to golf carts or electric bikes, but it would have been possible.

Because EVs didn't have the range of gasoline vehicles, EV concepts in the old days usually included something like overhead catenary power lines, or in-road induction or other "3rd rail" type charging on freeways, in order to sustain longer trips.

Let's try to keep in-mind that 1960s cars were horrendously unreliable, not starting when you turned the key because they were too hot or too cold, requiring obscene amounts of maintenance, etc., so a trade-off of range in exchange for an immensely reliable vehicle is something many people would have been willing to make, too.