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.
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.