Translation:I walk out of the bank and I walk into the hospital.
Out of the bank - you don't say "out from the bank" I'm pretty sure. "From the bank" means that we walk away from the bank building or there are 3 miles from the bank to the next street...
Would the sentence mean the same if it just said sétálok instead of besétálok?
Magyars love the verbal prefix. If you say "sétálok a kórházba" insted of "besétálok a kórházba", then it means slightly different: you are somewhere else, so you're walking towards the hospital.