The blaze took hold in the tower block in North Kensington, west London, shortly before 1am, meaning most people would have been asleep at the time.
Nearby St Clement’s Church was offering spare clothes and toiletries to anyone who had escaped wearing only their pyjamas.
The church appealed for people to donate any spare clothes or toiletries they have to help those who had come to their impromptu evacuation shelter.
Several other nearby community centres and places of worship were open to offer refuge.
Clement Attlee Estate Hall, at 21 Len Freeman Place, Central Gurduara Sikh Temple on 62 Queensdale Rd and The Rugby Portobello Trust at 221 Walmer Road were among those offering help and shelter.
Local Muslims were also reportedly on hand to help those who were evacuated from the tower as many were awake for Suhoor, a dawn meal that takes place during Ramadan.
A local mosque was also offering shelter and assistance to those affected.
A lot of people simply offered a cup of tea and a place to rest for people who had been forced to leave their homes.
Telecoms company Talk Talk, which has an office just down the road from the tower, was among those offering shelter.
Phone company EE also tweeted that anyone who needed shelter or to use a phone should visit one of their nearby shops in Kensington High Street, Notting Hill, Queensway, and White City.
As ash and debris filled the air in the area surrounding the smouldering tower, a local shop handed out dust masks to protect those who were evacuated.
The British Red Cross, whose volunteers were on the scene, praised the sense of community spirit Kensington residents had show in rallying round to help those whose homes had been devastated by the fire.
“Nine volunteers from the British Red Cross are providing practical and emotional support, including breakfast items, at a rest centre for residents affected by the terrible fire in Chelsea this morning,” Jon Pewtner, senior emergency response officer for the British Red Cross said in a statement.
“More are on standby to provide further support to emergency services throughout the day.
“The atmosphere here is tense – people are worried about loved ones and many are coming to the realisation that they have lost their homes.
“The community is rallying round, with residents collecting food, drink and clothes, and everyone is asking each other if they can help, or if they need food.
“There is a good, strong community spirit.”