Like all cars, Corvettes have fuel tanks - and vintage Corvettes have steel fuel tanks that are subject to rust and abrasion over the years. Many older Corvettes have had their tanks replaced at least once since they were new, and many more are in need of replacement.
Even more often, the flexible fuel lines that connect the fuel tank to the hard steel fuel lines that run along the car's frame are likely to be rotten in cars over 20 years old. These lines often seep and weep fuel long before they burst open and leak gas all over the floor - so if you've been smelling gas around your Corvette, chances are good that your fuel lines are cracked.
Fortunately, replacing your vintage third generation Corvette's fuel tank (1968-1982) is something you can do at home if you're reasonably handy. The precise instructions will differ from model year to model year, so as always be sure to get yourself a good repair and assembly manual for your particular Corvette. But the basics are all the same, so read this article and then you can decide if this is a task that you want to undertake.
NOTE: The procedure is likely to be similar for C1 (1953-1962), C2(1963-1967), and C4 (1984-1996) Corvettes, but I didn't have access to these to test these instructions. So if you have one of those models, you'll really need to depend on your repair manual.