There are some necessary governmental functions and total gridlock would be unfortunate. Some legislation is needed, for example
selective defense appropriations, anti-terrorism appropriations, opening up oil drilling lands and waters and speeding up nuclear power generation.
Here is a suggested procedure for killing ObamaCare, EPA regulations on carbon emissions and much of the other stuff dependent on future appropriations -- without producing gridlock -- in a way the Senate would find very difficult to block and President Obama would find very difficult to veto.
Should anyone take the time to read the article, please also see Comment #21. It links an article which sums up the possibilities, albeit from the standpoint of how horrible it would be not to fund ObamaCare. Reading it induced uncontrollable fits of giggling and made me wonder whether that might be covered under ObamaCare even though it is, sadly, a pre-existing condition.