Answering services, as known in the 1960s through the 1980s, earlier and slightly later, involved a business that specifically provided the service.
Primarily by the use of an off-premises extension (OPX) for each subscribing business, connected at a switchboard at the answering service business, the answering service would answer the otherwise unattended phones of the subscribing businesses with a live operator.
The live operator could take messages or relay information, doing so with greater human interactivity than a mechanical answering machine. Although undoubtedly more costly (the human service, the cost of setting up and paying the phone company for the OPX on a monthly basis), it had the advantage of being more ready to respond to the unique needs of after-hours callers.
The answering service operators also had the option of calling the client and alerting them to particularly important calls. The origins of call centers dates back to the 1960s with the UK-based Birmingham Press and Mail, which installed Private Automated Business Exchanges (PABX) to have rows of agents handling customer contacts.