1-2 hours would be best.
2 hours and you can thoroughly go through the exhibits. Old Courthouse and the Arch aren't terribly far along with the Cardinals Hall of Fame and Main Branch of the St. Louis Public Library which sometimes have special exhibits.
The City Museum is close and is very different, a great place for kids.
30 mins to 2 hrs
Gateway Memorial Museum at the Arch. Old Courthouse, Old Cathedral, Baseball Hall of Fame by the Stadium are all great places to visit and walking distance from each other.
If you take the time to read the exhibits, and watch the documentary, I'd say 2 hours unrushed.
If you want to go through the entire facility including the outdoor section, at least an hour maybe more. Gateway Arch is at riverfront, 12 blocks away. Part of the Jefferson Expansion Memorial which includes the Old Courthouse where a slave sued for his freedom. The home of the St. Louis Cardinals is nearby and while you can't get inside, it's a great stadium from the outside.
You can tour the museum in about an hour. The memorial is in close proximity of other landmarks and points of interest such as the recently-renovated Gateway Arch museum, Courthouse, and Old Cathedral.
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