The Appalachian Trail is in the United States and traverses over 2,000 miles through 14 states. It takes an averages of 5-6 months to complete. Hikers camp out along the way or stay in shelters on the trail. They can resupply on food and shower in trail towns a couple times a week. Hundreds of people complete this trail every year.
One of the coolest things about hiking in Nepal is the tea house system. During the day, you can hike among the tallest mountains in the world. At night, you can stay at tea houses that are owned by locals. It's a great way to meet both people from Nepal and other hikers like yourself. Not to mention, you don't have to carry camping gear during your journey which will help you during those high altitude treks.
Peru's Cordillera Huayhuash Circuit is a tough trail that takes about 10 days to complete. You will be camping the entire time and hiking over high-altitude passes. There are alpine lakes, snow-capped mountains, mule trains that pass you by, and one campsite even has a hot spring to soak in. This is a difficult hike and not yet crowded by tourists. If you're looking for a remote and secluded mountain experience, the Cordillera Huayhuash is for you.
It sounds like Eastern Europe is the place for you. Day hiking options are endless! You can stay with locals in guest houses in Albania's Accursed Mountains. Or you can rent a car and explore Montenegro's Durmitor National Park. Or you can take local buses to trailheads in Romania. Eastern Europe is an affordable hiking destination while still getting to sleep indoors.