On a world map, the 180th meridian divides the Eastern Hemisphere from the Western Hemisphere. The area it passes through is mostly water, but remote parts of Russia, Alaska, Fiji, and a few others pass near or through it.
Is there any country I can set foot on both hemispheres as a traveler if I wanted to? If so where can I visit to do so?
Yes, on Taveuni Island, the third-largest island in Fiji, and a popular tourist destination.
The 180 meridian is crossed by a road. This image is from Google Earth
There is a nearby hotel called First Light Inn.
In Chukotka, far eastern Russia, there is a monument where the 180° meridian and the Arctic Circle meet, though getting there is apparently something of a trek; as the linked website states:
If the weather is fine, you can reach the meridian by car after about a 60-km ride along the Egvekinot — Valunisty gold mine road. There is a sign post by the road. However, this still is not the 180th meridian. To get to the intersection point, you will need to walk about 2 km more in the direction indicated by an arrow laid in stone.
There also appears to be a road, though it might be unpaved and/or accessible during only part of the year, running near the Arctic coast which crosses the 180° meridian a bit west of Mys Shmidta. Wikipedia references suggest that there is some information about this road in Petit Futé Chukotka if you can somehow track down a copy.
Chukotka is a very remote and harsh part of the world; a special permit from the Russian government is required to visit, and you probably want to engage the services of a qualified guide in order not to die from the cold weather and limited services.
Clearly Antarctica. Either stroll around it at Amundsen-Scott South Pole Station (US), as many do, or take a skidoo from McMurdo or Scott Base (NZ) a little ways across the Ross Ice Shelf.
