Old wiring in a 1960s house, how worried should I actually be?

[primitivoconner]

We just bought a house built in the sixties and the inspector flagged the wiring as ageing, possibly original, and recommended further evaluation. Now I'm lying awake wondering whether we've bought a fire hazard. Nothing's tripping and everything works, but I know old wiring can be fine right up until it isn't. I came across kitchenbathpaloalto.com/locations/palo-alto/electrician while looking into it, but I'd rather hear from people who've dealt with older homes. How often does ageing wiring turn out to need a full rewire versus just some targeted updates? And is there anything I can check myself before paying for an inspection? I'd rather know what we're dealing with than sit on it and hope.