Dean McIlwaine, from Newtownabbey, was last seen in the Carnmoney Road area of Newtownabbey on Thursday, 13 July.
Hundreds of people joined a search in Belfast on Monday night in a bid to find the 22-year-old barber.
His cousin, Jordan Malone, told the BBC that the family has barely slept since Mr McIlwaine's disappearance five days ago.
He said that he was very close with Mr McIlwaine, as they are both barbers - and that the family cannot understand why he would suddenly go missing.
"Despite the search last night, we've still heard nothing - we're just going up against a brick wall," said Mr Malone.
"Dean and I talked every day, he would never do something like this. Someone would normally know where he is."
The police said Mr McIlwaine was last spotted on the Carnmoney Road on 13 July at about 13:00 BST, wearing a light coloured t-shirt, blue denim shorts and flip-flops.
They have been carrying out searches in the Carnmoney area and across Belfast and renewed their appeal on Tuesday evening.
Hundreds of people, including Mr McIlwaine's father and brother, also took part in a search in Belfast on Monday evening, organised by the Belfast City Council.
Mr Malone said his aunt and uncle have barely slept since their son's disappearance, and are finding it incredibly difficult to cope.
"I think they've had three hours of sleep between them since last Thursday, his brother is like a sheet - he can't take it in," added Mr Malone.
"His mum can't speak. Every time the phone rings or the police arrive, she's afraid it's going to be the worst news."
Mr McIlwaine is described as being 5ft 8 ins tall, of medium build with brown eyes, dark hair and a beard.
He has sleeve tattoos on both arms.
Mr Malone said the family had taken comfort in the support they have received from people since his cousin vanished, and added that they are "so determined" to keep searching for him.
"We'll not give up, we'll keep searching, but as the days go on it does look more negative - there is just no trace of him," he added.
He also had a poignant message for his cousin, in the hope that Mr McIlwaine eventually makes contact with the family.
"I just want to say to him no-one is angry or mad at you. Whatever is going through your head we can help you, we just need to know you're safe.
"We just need to hear from you."
The police have asked anyone who may have seen Mr McIlwaine to contact them on the non-emergency number 101, quoting reference number 121 of 14/7/17.
The family of a missing County Antrim man have said they are determined to "keep searching" until he is found.