GB were beaten 24-18 by Israel, 37-14 by Hungary and 46-14 by Netherlands.
The men's and women's teams had their £2.92m funding to support elite athletes removed after London 2012.
"Although we did not make the finals, there are some great young handball players in this country," said British Handball director David Meli.
"Our aim now is to find a way of bringing additional funding into the association to allow us to run other teams in both male and female competitions."
Netherlands won all three games to secure their place at the finals in Rio, which take place from 19 July to 2 August.

The proposed tourist development would also include a visitors' centre and hotel, with a restaurant overlooking the water.
The Humber Bridge Board said it hoped to submit a planning application by the end of the month.
If approved the new centre could be open at the end of 2017.
The proposal would see visitors travelling under the bridge in a glass gondola before changing to a glass lift to ride to a platform on the top of the North tower.
Construction began in July 1972, taking eight years to complete
It is held up by 44,117 miles (71,000km) of steel wire, almost enough to travel twice around the world
The bridge is 7,283 ft (2,220 m) long from shore to shore
It weighs more than 500,000 tonnes
With a central span of 4,626 ft (1,410m), it is the seventh longest suspension bridge in the world
It was officially opened by HM the Queen on 17 July 1981
Source: Humber Bridge Board
Bridge Master Peter Hill said the scheme was about "making the best of what we've got".
"People have for years had wanted to get more access to their bridge, so we're going to make it happen, " he said.
Mr Hill said the project would be financed by the bridge board but "would not be funded from bridge tolls".
A public exhibition of the proposals has gone on display at the bridge.

Broken swords and spearheads were found by archaeologists on the RSPB Scotland nature reserve.
Twelve pieces excavated from several different weapons have been handed over to Kilmartin Museum in Argyll.
RSPB Scotland reserves archaeologist Jill Harden said they had probably been deliberately broken and thrown into a loch as part of a religious ceremony.
"This is the first discovery of this size from Argyll for many years," she said.
"The items were recovered from what had once been a freshwater loch - it seems that they had been purposely broken and cast into the waters as part of a ceremony, most likely as offerings or gifts to the gods or goddesses of the time.
"It is recorded that bronze swords were found on Coll in the 19th Century during drainage works, but their whereabouts today are unknown."
The archaeological investigation was directed by the Treasure Trove Unit, National Museums Scotland and RSPB Scotland.
Trevor Cowie, from National Museums Scotland's department of Scottish history and archaeology, said: "While a fair number of objects from this period have been discovered in the west of Scotland in the past, we generally know very little about the precise places where they were found.
"Archaeological techniques have developed dramatically since those 19th Century discoveries were made, so we have a great opportunity here to resolve many unanswered questions about life on Coll some 3,000 years ago."
The weapons can be viewed at the the Isle of Coll's An Cridhe community centre on Thursday and Friday.
Great Britain failed to qualify for the men's Under-21 World Handball Championship after losing all three qualifying games at Kent's Medway Park.
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Plans have been unveiled for a glass lift to take tourists to the top of one of Humber Bridge's 510 ft (155m) high towers.
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Bronze weapons believed to date back about 3,000 years have been discovered on the Isle of Coll.