It is definitely possible to join the medical field after one completes a BSc, although the route one may take depends on what specific area of medicine that particular student would wish to pursue. A student after BSc in fields like biology, biochemistry, or even medical sciences, can choose several courses and master science programs relating to medical, or instead choose for a medical degree. Some students continue towards a Bachelor of Medicine, Bachelor of Surgery or MBBS and appear for the entrance exams to get into the medical fields, based on their interests or career aspirations, but MBBS intake is highly competitive with a lot of dedication and effort.
Or, if you prefer the allied health field, then many other paths will open, for example, being a medical laboratory technologist, a radiographer, or physiotherapist and being a nurse. Each has a prerequisite in additional certification and diploma for certain areas of specialties. Put it this way: BSc could act as an appropriate point of takeoff into medicine and health; then, based on your desired specialisation in that aspect of medicine, you may make further specific decisions regarding a particular subsequent level or path to undertake.