In the US, referendums are handled on a statewide basis. There's no constitutional basis for having a national one. So if you wanted to effect a national referendum, you'd need to get the same referendum put up in all 50 states for the same election.
That may sound a bit daunting, but candidates for president (at least in the primaries), have to go through roughly the same process. There's no automatic way to get put on every state ballot, so supporters must get a primary candidate on the ballot individually in all 50 states through each one of those state's process for that.
Of course there's nothing the individual states can decide this way that is binding on the Federal Government. So all you'd end up with by doing this is an overly thorough publicly-financed opinion poll. Still, this is often done in individual states as a way to encourage certain voters to come out to vote in a particular election.
The exception here is a Constitutional Amendment. It is possible for one of those to get ratified by actions of the states, which could theoretically (if each state wanted to do it that way) come via a popular vote of that state. However, the deciding factor would be getting ratification of 3/4ths (right now 38) of the States. That could in theory happen even if a majority of the voters across the country voted it down.