I have family members with gout. Does this mean I'm likely to get it?

May be. If you have positive family history of gout, that consider a risk factor you may devoloped gout and early.
You may be at risk. There are a variety of reasons people get gout. Gout is a symptom and you have to evaluate why the person has gout. There are some genetic changes that run in families that can lead to gout. See what type of gout your family members have and this suggests whether you are at risk. Some people make too much uric acid which can be genetic while others do not eliminate the uric acid.