I think it's a role that a lot of actors could do justice actually. I'd prefer if they stuck with the original Indiana Jones formula and cast an actor in his late 30's or 40's - remember the guy is a tenured professor and has spent years in the field. To that end I think someone like Clive Owen, Damon, McConaughey, Fassbender, DiCaprio, Pitt, Jackman, Butler, Renner, etc.
Knowing Hollywood they'd probably do a reboot showing Indy in university like that piece of crap Jack Ryan: Shadow Recruit, in which case Gosling or Chris Evans are about the only young guys that I can see playing the role well.
Sean Patrick Flanery, the guy who played Indy in the Young Indiana Jones Chronicles back in the 90's, did a great job I think, and is the right age to play Indy now, but never in a million years would Hollywood cast him as the lead in a $100 million + blockbuster. He's still active in Hollywood (he was awesome in Boondock Saints, and I've seen him more recently in Criminal Minds and Dexter).
edit: just looked him up on IMDB and he's 48! Jesus.
For the love of god, please no more LeBeouf.