Hi, I was in Nashville for UG and a couple of years after, and during this time I knew a handful of students who didn't have a car. However, all of them had access to a car and friends (like myself) willing to schlepp people to and fro. The Hillsboro village/ 21st avenue area is your best best in terms of access to laundromats, grocery stores, coffee shops, bars, restaurants and even a movie theatre. However, if you're interested in going to East Nashville (the 'hip' area of Nashville) or the 12th South area (lots of good restaurants) you will need a car. Personally, I would make it a priority to have a car if at all possible, though parking at Vanderbilt is horrific.
A walk from the hillsboro area to Vandy's campus ( in particular the english dept bldgs) would take you maybe 20-30 minutes - not a bad walk at all. As far as rents go in that area, I have a friend who had an amazing 1 bedroom for probably $800/month. I've lived in that area, and I would say that you are probably going to pay around $500/mo minimum for a room in a 2bd or 3bd. If you want to live alone, I would say you're looking $800 minimum although I'm not really sure about that. In short, I think Nashville wants to be a small progressive city that is walkable and bikeable but it's not there just yet. It's a great place to live,though, and feel free to PM me with any questions about Vandy or Nashville.