Can I still pursue a research career after a possibly lengthy break? I've been offered a PhD position, but I'm forced to decline it due to family reasons. However, once the situation becomes more stable, I will re-apply again. What concerns me most is whether it's still possible for me to pursue a research career after a long (2-4 year) break?

I hope that perhaps once the crisis is overcome, I could look for part-time assistant/technician jobs in labs, which would allow me to still stay close in research before I would submit my applications for a PhD.
Please contact the dean of graduate studies at the department that accepted you and describe your situation.  Various creative approaches to this problem may be possible.  For example, if the first stage of your degree program involves coursework to prepare for some qualifying exams, it may be possible for you to take some of these courses at a university closer to home, with careful planning, and direct communication between the two departments.
If you are not already working with a counselor or therapist, I suggest that you find someone you like to get started with.  When going through a family upheaval this can be helpful.  Also, in the safety of the counseling relationship, you may gradually become comfortable about exploring different ways of balancing supporting your family and pursuing your own personal growth.  Sometimes the stress of the situation one is living through at the moment makes it difficult to see all possible options.