The Backpacker's Inn is a small but very nice little hostel located right in downtown East Glacier Park. If traveling by train, it will be behind you when you get off at the station. Just walk left, under the railroad tracks (this is State Highway 49) and left again into downtown. This is US Highway 2. Look for signs for Serrano's Mexican Restaurant. It is just off the highway and very easy to find.
The hostel is in the large and very pleasant backyard of the restaurant. The grassy space contains a picnic table, flowers, several trees, a tire swing, and two dogs. There is a row of four simple, clean rooms, each perhaps twelve feet by twelve feet in size. We did not go in the two private rooms, but the two dorm rooms -- one each for men and women -- each contain two bunk beds with sufficient foam pads and fitted sheets, a sink (with signs noting a "Boil Water Advisory"), a mirror, a topographical map of the park, and a few pictures. It's sparse but clean, and the building looks like it's in good shape.
There's no common room -- only the large backyard -- and the small size of the place contributes to a friendly atmosphere among travelers, who are always good for tips about the area. Most people are indeed backpackers just going or coming from the nearby trails into the park.
There is no place to store bags securely, although the rooms are fairly secluded and the town is very small. There are no linens and there is only one bathroom, located in the men's room. There is a door between the rooms that is lockable on the women's side only. Presumably the single rooms have their own bathrooms.
Despite what this website says, you may want to call them about reservations further in advance, especially if you are thinking about trying to get a single room. Upon calling the day before our desired arrival, the two private rooms were already taken. The restaurant, for which there are menus in all the rooms, is open from 5 p.m. to 10 p.m. daily, although people are there throughout the day. The staff in the restaurant doesn't always answer the phone right away during non-business hours, so just call back often if they don't answer.
As well as being within a stone's throw of the train station, the Backpacker's Inn is just across the street from the town's best grocery store. Also, Brownie's, which is about a half-mile walk away, allows anyone to use their internet computers for a price. The nearest trailheads entering Glacier National Park are about a mile away (Brownie's is a bit closer to them). Just start going north take a left on First Street and continue down the gravel road to one or two trailheads (including the Mt. Henry trail). Another trail may be off a different gravel road in the same general area, according to the map at the hostel.
In general, you can't beat the price and location.