The swap file is a kind of "memory extension". During the runtime of the operating system, Windows write there its data not needed at the moment (to free the physical memory for other data that are needed at the moment). When you shutdown your computer, the swap file content is discarded (it's never used anymore, even though the file may stay as big as it was at the end).
If something fills your swap file - it may mean that there is a memory leak in one of your applications. You may try to run some task manager tool to find out if any of your running processes is eating an extraordinary amount of memory. However, I wouldn't blame a virus for it (especially if avast! didn't find it; and no, the viruses cannot "escape" from the Chest).
One more thing: Windows 9x has a strange (politely said!) default memory management behavior. To make the system usable, you should tune the settings in system.ini. In particular, you should limit the maximum size of Windows cache ([vcache] section) and put the ConservativeSwapFileUsage=1 value to [386Enh] section. I'm sure google has a number of references for this.