Sir Keir said he backed Brexit Secretary David Davis' previous pledge that Britain should not lose single market and customs union benefits when the UK quits the Brussels bloc.
And appearing on The Andrew Marr Show, Mr Starmer calmly stated the Tory minister must now uphold that.
The Labour Party will tomorrow set out six tests for the final deal for Brexit and Mr Starmer has said Labour will not support any Brexit deal unless these tests are met.
Mr Marr said: “You say we want the exact same benefits, your words, from the single market and the customs union. Can I put to you that that is just not going to happen, we’re outside both of those things.
“Michel Barnier and others made it absolutely plain that we cannot get all the same benefits as if we were inside and therefore that is a hopeless thing to ask for.”
Keir Starmer said the UK must gain the "exact same benefits" from the EU once it leaves the bloc
If we crash out without a deal, we’ve no meaningful relationship with our EU partners
But the MP for Holborn and St Pancras said the Government must now achieve this aim after Mr Davis had said just that.
Sir Keir said: “We’re holding him to that test. The Government can’t turn around now and say ‘this is unachievable’ because it was David Davis who the Prime Minister’s appointed in the role of exiting the EU, who set that.”
He said it would be “very damaging” for the UK not to secure a deal with the EU, adding: “If we crash out without a deal, we’ve no meaningful relationship with our EU partners.
Brexit: Results of how the UK voted Mon, March 20, 2017
Much of the North East of the United Kingdom voted to leave the European Union including Sunderland, Gateshead, Darlington, Durham, Hartlepool, Middlesbrough, Stockton, Redcar and Cleveland, North Tyneside and South Tyneside, and Northumberland
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GREAT YARMOUTH: The town of Great Yarmouth on the East Coast of England voted by 72% to leave the European Union.
“Now I accept we won’t be members, but we must be partners and this has to be fought for over the next two years.”