Mike's comment is appropriate if you're talking about using an Uno or other '328 class machine as they only have 2K of ram. As basic is usually interpreted, programs would normally be loaded into ram and since ASCII text is one byte per character, that leaves little room for the ram needs of the interpreter and the actual user program.
Yes, of course a fixed, proven and non-changing program like adventure could be placed into the flash space and run from there but the interpreter would have to implement that as well as being able to run in interactive mode from ram if that was a goal.
An ATmega1284P would give you 16k of ram. Or there's always a Teensy which has 256k in the 3.6 variant which is a ~$30 board. In the end, there are lots of choices that could work.