BF is turing complete. It also has a way of doing input and output. That means yes, it is possible.
It would be horribly slow though, and it would be complete hell to code. Even just doing multiplication was a real brain twister.
For the record, here's a piece of BF code to run multiplication I made during a boring class at school:
,>++++++++[<------>-],>++++++++[<------>-]<[>+>+<<-]>>[-<<+>>]<<<[>>><[>>+<<-]<[>+>+<<-]>>[-<<+>>]<<<-]>>>>[-<<<<+>>>>]<<[-<->]<<.
It would be horribly slow though, and it would be complete hell to code. Even just doing multiplication was a real brain twister.
For the record, here's a piece of BF code to run multiplication I made during a boring class at school:
,>++++++++[<------>-],>++++++++[<------>-]<[>+>+<<-]>>[-<<+>>]<<<[>>><[>>+<<-]<[>+>+<<-]>>[-<<+>>]<<<-]>>>>[-<<<<+>>>>]<<[-<->]<<.