So you got me thinking; I think I may add one myself as an NPC.
I'm going to start with a human with the Archaeologist background from TOA, NG.
I don't want him to have magic, and he doesn't need to be a frontline battler so I think I'll go with Rogue.
For weapons a whip and a revolver (or pistol p. 268 DMG; or just hand crossbow if your DM won't allow it).
Skills would be: insight, investigation, perception, persuasion.
For Expertise, he'll have Wisdom (perception to check for traps) and thieves' tools.
For archetype, probably just thief.
From a purely narrative and character perspective - Indiana Jones probably is a Ranger. Why? A Ranger is the expert in surviving a certain environment or location - he's the guy who, when you ask who you need to bring along with you who knows what's going on, you call first. .
Issue is for whip he needs a feat or a class with martial weapons to use one so he would have to add some fighter be the easiest and Human variant and take tavern Brawler because Indy got into a lot of fist fights
OMFG Henry Jones Sr is a deeply religious person on a holy quest. Cleric w/sage background seems fitting.
I'm the guy who thinks Indy could use a few levels of Monk for his fisticuffs and unarmored AC. Basic ki usage like disengage or dash and dodge as well as increased movement (he books from people chasing him all the time).
If you REALLY want to, I guess you could take Sun Soul monk from SCAG and just reskin the radiant sun bolts as pistol shots. But rogue seems better than more monk levels.
There's been talk about variant human, if not needed for Tavern Brawler due to monk I think the Dungeon Delver feat is a great fit for Indy.