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Fourth place could provide European football if one of the Premiership's top three win the Scottish Cup.
But their 0-0 draw at Kilmarnock leaves Hearts 10 points behind Rangers and four behind fourth-placed St Johnstone, who both have a game in hand.
"Fourth place is 100% our aim and ambition," Smith told BBC Scotland.
"We play St Johnstone and the other top teams in the division so points can be picked up and dropped anywhere. If we can keep within reach of St Johnstone until that game then it's massive for us.
"If we can achieve European football then it's the bare minimum I think Hearts should be achieving.
"It's frustrating for us coming away and not getting the three points. We maybe didn't play as well as we wanted to or create as many chances as we wanted to, but we could have nicked it.
"Maybe six weeks ago we would have come here and not got a clean sheet and maybe lost the game 1-0."
Hearts were second in the Premiership when Robbie Neilson left in early December, but have won only five - and lost nine - of their 17 league games since Ian Cathro took over as head coach.
"It's been a big turnaround since Robbie Neilson left, with new ideas and new faces, and it can only be good for me as a young player dealing with all these different things," Smith added.
Hearts full-back Liam Smith says a Europa League qualifying spot should be the "bare minimum" requirement for the Tynecastle club.