I haven't had a car through undergrad and my Master's and will not be buying one during my PhD. I'm 39 and have two teenage daughters. We walk, take the bus, and, in the summer (we get snow in the winter), ride our bikes. Yes, there have been times I missed having a car, but I'm actually proud not to own one. Resist the culture of car-dependency and encourage more sustainable urban planning! (From today's NY Times, "Europe Stifles Drivers in Favor of Mass Transit and Walking" - http://www.nytimes.com/2011/06/27/science/earth/27traffic.html?_r=1&nl=todaysheadlines&emc=tha2)
Not owning a car is much cheaper (a car costs money even sitting in the driveway, let alone the rising cost of fuel, repairs and insurance; plus not having a car reduces those impulse outtings or trips to the store) and much better for the environment and for me - more exercise! Even with a mediocre public transit system where I live and two kids to transport and run errands for, I just can't justify the expense of a car. It may be less convenient and take a bit more planning at times, but for me it's worth the savings. I can put the money to better use elsewhere.
Having said that, I'm not at all familiar with Nashville, and only you can decide how much running around you will need to do and how much of it can be done on foot or bike or by bus. I agree with consulting current students in your program. If you are concerned about the bulk and weight of groceries, shop for bulky and heavy things only once every other week or once a month (stock up!) and splurge on a taxi to get home. Still cheaper than owning a car. Coordinate with your roommate and you can split the cost of the taxi.