Yes. Absolutely yes. My first group had an Elven barbarian and none of us were bothered by that; we thought it was cool. What I will say is that the class has to match the character; we wouldn't have accepted that elf had she been really prissy or bookish the rest of the time.
Also, there has to be an in-character reason for why they have that class. My group is fairly easygoing so there's never any trouble with things like dwarf wizards, so long as they can tell us when they learned that class.
One thing I will say is, for any area which might be considered a point of contention within the group, talk to your GM. If he/she's any kind of decent as a GM then, unless there's an in-universe reason why that cannot exist (our first campaign had no orcs/half-orcs allowed for the players because all the orcs had been genetically modified into being Warhammer orcs by the big bad and my first campaign had no elves/half-elves due to them not existing, but in these cases the GM should tell you before you make your character), then they'll work with you to figure out a way that that would work.
Also, there has to be an in-character reason for why they have that class. My group is fairly easygoing so there's never any trouble with things like dwarf wizards, so long as they can tell us when they learned that class.
One thing I will say is, for any area which might be considered a point of contention within the group, talk to your GM. If he/she's any kind of decent as a GM then, unless there's an in-universe reason why that cannot exist (our first campaign had no orcs/half-orcs allowed for the players because all the orcs had been genetically modified into being Warhammer orcs by the big bad and my first campaign had no elves/half-elves due to them not existing, but in these cases the GM should tell you before you make your character), then they'll work with you to figure out a way that that would work.