The workers have been sent home and Royal Mail said an intensive clean of the building has been carried out.
The office is operating ‘as normal’, according to the postal service, despite the four positive cases.
A spokesman for Royal Mail said: “Royal Mail takes the health and safety of its colleagues, its customers and the local communities in which we operate very seriously.
“Four of our colleagues recently tested positive for coronavirus at the Hastings Delivery Office. They are now recuperating at home and we wish them a speedy recovery.
“We have carried out an intensive clean of the building. The office is operating as normal.”
This is the second incident involving Covid-19 at the Hastings delivery office.
In July, Royal Mail confirmed one member of staff had tested positive and a deep clean of the office was carried out.
On Thursday (October 8) Royal Mail outlined the measures that have been in place throughout the Covid-19 pandemic to keep customers and colleagues safe.
The spokesman added: “Throughout this crisis, every decision we make puts the health of our people and customers first.
“We have put in place a range of preventive measures to protect both our customers and our colleagues.
“We were the first UK company to put in place social-distancing measures in relation to parcel delivery. We pioneered contact-free delivery. We are temporarily not handing over our hand-held devices to customers to capture signatures. As well as encouraging good hand hygiene, standard ways of working have been revised to ensure that colleagues stay two metres apart at all times.
“In addition we have moved to a one person per van delivery model. All staff have been briefed about the social distancing measures jointly agreed by local management and the CWU. This has been supplemented with visible reminders such as posters and one-way floor markings.”