The fair place to meet is Columbus, OH — the city closest to the midpoint of Indianapolis and Washington. From the farther side that’s about 6 hr 51 min of driving.
Recommended midpoint
Columbus, OH
From Indianapolis
3 hr 31 min
168 mi to Columbus
From Washington
6 hr 51 min
326 mi to Columbus
Indianapolis has the shorter trip; the split is off by about 3 hr 20 min. The alternatives below can even it out. Indianapolis and Washington are about 491 miles apart.
Indianapolis and Washington are about 491 miles apart by road. Split the difference and you arrive near Columbus, the city closest to the halfway point between them. That puts roughly 3 hr 31 min of driving on the Indianapolis side and 6 hr 51 min on the Washington side — the fairest single meeting point among the cities near the middle.
That's a half-day drive from each side, so Columbus suits an overnight or a weekend rather than a quick coffee — long enough to want a reason to stay, short enough to drive.
If Columbus doesn't have what you're after, Pittsburgh and Cleveland are also close to the midpoint and worth a look — each keeps the drive reasonably balanced between Indianapolis and Washington.
Columbus is the capital and most populous city of the U.S. state of Ohio. With a population of 905,748 at the 2020 census, it is the 14th-most populous city in the U.S., second-most populous city in the Midwest, and third-most populous U.S. state capital. Read more →
Indianapolis, colloquially known as Indy, is the capital and most populous city of the U.S. state of Indiana and the county seat of Marion County. Indianapolis is situated in the state's central till plain region along the west fork of the White River. Read more →
Washington, D.C., officially the District of Columbia and commonly known as simply Washington or D.C., is the capital city and federal district of the United States. Read more →
City descriptions adapted from Wikipedia (CC BY-SA); photos via Wikimedia Commons, credited above.
Estimates use straight-line distance and typical road speeds; real drive times vary with route and traffic.