Is it reasonable to go to Longyearbyen by myself without a car?

As others have said, a car is not needed and not that useful. But there are in fact a few roads and a taxi service beside the airport shuttle. Nybyen, where the only hostel is located, is a bit outside the rest of Longyearbyen, maybe a couple of kilometres up Longyear Valley. That's perfectly doable by foot (that's what I did most of the time last winter and most people do the same) but getting a taxi is also an option (and at the end of the day probably cheaper than hiring a car, even if you make two-three return trips a day).

The thing is that getting out of the town without a guide – even on the road along Adventdalen – is not recommended, due to the presence of polar bears. People have died hiking on the mountains immediately next to Longyearbyen without proper equipment (gun, etc.) and a bear and her two cubs walked right in the middle of town last winter.

So even if you go somewhere reachable by land (visit to mine 3, snowmobile trips – even in summer, hikes, drive up Adventdalen, etc.), you should hire a guide and they will arrange transfer (pick you up from your accomodation and drive as far as the road goes). Car or no car, that's what you'll need to do if you want to do anything else than walk around the town and see the museum.

Apart from the airport bus, Longyearbyen has no public Transport, but is small enough for you to walk from end to end.