BBC BASIC is not immune from file size limitations; by default only 7936 bytes (&A100-&C000) are available to BBC BASIC. Typically BBC BASIC would use external storage to save files to and load them from (such a a floppy or tape drive). As the TI-83+ stores everything in RAM you could have two copies of a program in RAM at once; one in BBC BASIC's memory and the other in TI-OS memory (saved as a variable).
You can give BBC BASIC more memory by raising HIMEM with
*HIMEM, but this eats into space available for saving files into. Ideally the BBC BASIC programs would be written externally (and I suppose it would be possible to write an editor that ran on the calculator outside BBC BASIC itself) then archive them; BBC BASIC can then load these files into its own RAM directly from the archive.
For a real-world figure, a TI-83+ with 24312 bytes free RAM can have 21389 given to BBC BASIC. This memory would be shared between the program code, variables (heap) and BASIC stack, but isn't
too shabby. It's certainly more that some of the original BBC Micros had!
