The French heavyweights, twice Heineken Cup runners-up, stand between Exeter and a place in the competition's penultimate knockout round.
And current form suggests that Exeter - fifth in the Aviva Premiership and pushing strongly for a play-off place - can head to Paris on Thursday confident of testing Stade.
"This is the biggest game the club has ever played - we've never been at this level before," Hayes said.
"There was a lot of euphoria attached to our promotion wins against Bristol, and to certain wins in the Premiership, but when you look at it the Amlin Challenge Cup is probably the highest level of competition the club has ever played in.
"This is the rugby we all want to be playing in, and there is no point in us being afraid of being here. We have to try and enjoy the challenge.
"We realise they've got some massive names and Stade has been a marquee name in rugby for 10 or 15 years now, but they are the big challenge that we have to relish and we have to face up to."
Head coach Rob Baxter has retained the team that accounted for London Irish in last Saturday's Premiership clash at Sandy Park, being able to do so after hooker Chris Whitehead recovered quickly from an ankle injury.
"Thankfully, everybody who we had concerns over have come through 100% fit," Baxter said.
"There are some teams who are probably already looking forward to their holidays, but we are not. Instead, we have a number of games left that we want to win, and this is the first of them.
"We've already had some real challenges this season, not least playing the likes of Leicester.
"We came through two games against them, and we also came through what was a very tough final quarter up at Gloucester.
"I have kind of said to the players that I hope it is tough over there because I want them to look at each other afterwards and say we have come through something special.
"There will be nerves pre-match, but that is to be expected, and when we are like that and we're a bit edgy we tend to focus more and become an even better side."
Stade breezed through their qualifying group, averaging 40 points a game, and they will field a side containing four players who saw RBS 6 Nations action this season in Sergio Parisse, Julien Dupuy, Dimitri Szarzewski and Pascal Pape.
Former England wing Paul Sackey also starts, along with two Australian imports in centre Morgan Turinui and flanker George Smith as Stade look to reach a second successive Challenge Cup semi-final.
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