There's no way out of this situation.
In a fire, it's important to secure an escape route.
There was no escape from his persistent questioning.
Having lost any way out, he finally told the truth.
People sought refuge from the flood by heading to higher ground.
避難所 is an officially designated evacuation shelter for disasters, while 逃げ場 is any place or means of escape, physical or figurative.
出口 refers to a literal physical exit, whereas 逃げ場 can be a physical place or a metaphorical way out of a difficult situation.
Standard spelling; common in news and everyday usage.
Less common variant without okurigana; still encountered.
Compound of 逃げ (にげ, 'escape', continuative form of 逃げる) + 場 (ば, 'place').