noun
turnout; lay-by; passing place; rest stop
A widened area on a narrow road or highway where vehicles can pull over to let others pass or take a short break.
山道には待避所がいくつか設けられている。
Several turnouts are provided along the mountain road.
対向車が来たので、待避所で待った。
An oncoming car approached, so I waited at the passing place.
noun
A place offering protection from danger, weather, or disaster; broader than the road-specific sense.
地震のときは、この建物が待避所になる。
In an earthquake, this building serves as a shelter.
登山者は嵐を避けるため待避所に駆け込んだ。
The climbers rushed into the refuge to escape the storm.
避難所 is the common word for an evacuation shelter during disasters; 待避所 is less common and can also refer to a roadside turnout.
Compound of 待避 (たいひ, 'waiting and taking refuge') + 所 (じょ, 'place'). The exact historical derivation is uncertain, but the word is formed from standard Sino-Japanese elements.